Chebyshev polynomials related to Jacobi weights
Jacob S. Christiansen Olof Rubin

	AdvSecureNet: A Python Toolkit for Adversarial Machine Learning
    Melih Catal melihcatal@gmail.com 

       Software Evolution and Architecture Lab 

       University of Zurich, Switzerland
       Manuel Günther guenther@ifi.uzh.ch 

       Artificial Intelligence and Machine Learning Group

       University of Zurich, Switzerland

    September 9, 2024
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§ ABSTRACT 
	We investigate Chebyshev polynomials corresponding to Jacobi weights and determine monotonicity properties of their related Widom factors. This complements work by Bernstein from 1930-31 where the asymptotical behavior of the related Chebyshev norms was established. As a part of the proof, we analyze a Bernstein-type inequality for Jacobi polynomials due to Chow et al. Our findings shed new light on the asymptotical uniform bounds of Jacobi polynomials. We also show a relation between weighted Chebyshev polynomials on the unit circle and Jacobi weighted Chebyshev polynomials on [-1,1]. This generalizes work by Lachance et al. In order to complete the picture we provide numerical experiments on the remaining cases that our proof does not cover.





Keywords Chebyshev polynomial, Widom factor, Jacobi weight, Jacobi polynomial, Bernstein-type inequality
	
Mathematics Subject Classification 41A50. 30C10. 33C45.




	


§ INTRODUCTION

In an extensive two part analysis of extremal polynomials on the unit interval, found in <cit.>, S. N. Bernstein investigates the asymptotic behavior of the quantity

    inf_a_1,…,a_nsup_x∈ [-1,1]w(x)|∏_k=1^n(x-a_k)|

as n→∞ for a variety of different conditions on the weight function w:[-1,1]→ [0,∞). If w is bounded and strictly positive on a set consisting of at least n points on [-1,1], there exists a unique set of nodes {a_1^∗,…,a_n^∗}, all situated in [-1,1], such that

    sup_x∈ [-1,1]w(x)|∏_k=1^n(x-a_k^∗)| =inf_a_1,…,a_nsup_x∈ [-1,1]w(x)|∏_k=1^n(x-a_k)|,

see for instance <cit.>. In other words, the infimum of (<ref>) can be replaced by a minimum.

The polynomial 

    T_n^w(x) := ∏_k=1^n(x-a_k^∗)

is the weighted Chebyshev polynomial of degree n corresponding to the weight function w and it is the unique monic polynomial of degree n minimizing (<ref>). Under the assumption that the weight function is continuous, the minimizer of (<ref>) can be characterized by the so-called alternation property. That is to say, a monic polynomial P of degree n is equal to T_n^w (and thus minimizes (<ref>)) if and only if there exists n+1 points -1≤ x_0<…<x_n≤ 1 such that

    P_n(x_j)w(x_j) = (-1)^n-jmax_x∈ [-1,1]w(x)|P(x)|.

This result – which is central to the study of best approximations on the real line – can be found in many works on approximation theory, see for instance <cit.>.

 The study of (<ref>) dates back to the work of P. L. Chebyshev 
 <cit.> who considered weight functions of the form w(x) = 1/P(x) where P is a polynomial which is strictly positive on the interval [-1,1]. Later A. A. Markov <cit.> extended these results to allow for weight functions of the form w(x) = 1/√(P(x)), where again P is a polynomial which is assumed to be strictly positive on [-1,1]. It should be stressed that both Chebyshev and Markov provided explicit formulas for the minimizer T_n^w in these cases in terms of the polynomial P. These formulas can be conveniently found in Achiezer's monograph <cit.>.

The considerations in <cit.> were different from those of Chebyshev and Markov as Bernstein considered the asymptotic behavior of (<ref>) as n→∞ for a broad family of weights, not restricting himself to continuous ones. In particular, writing f(n)∼ g(n) as n→∞ if f(n)/g(n)→ 1 as n→∞, he obtained the following result. 

	Let b_k∈ [-1,1] and s_k∈ for k=1,…,m, and let w_0:[-1,1]→ (0,∞) be a Riemann integrable function such that there exists a value M>1 for which 1/M<w_0(x)<M holds for every x∈ [-1,1]. Consider the weight function given by 
	
    w(x) = w_0(x)∏_k=1^m|x-b_k|^s_k.

	Under the above conditions, we have that 
	
    min_a_1,…,a_nsup_x∈ [-1,1]w(x)|∏_k=1^n(x-a_k)|∼ 2^1-nexp{1/π∫_-1^1log w(x)/√(1-x^2)dx}

	as n→∞.
	

This result is shown in two steps. Bernstein initially considers weight functions w_0 as in Theorem <ref> and finds that if c_1^∗,…,c_n^∗ are the unique nodes satisfying

    ∫_-1^1|∏_k=1^n(x-c_k^∗)|^2w_0(x)^2/√(1-x^2)dx = min_c_1,…,c_n∫_-1^1|∏_k=1^n(x-c_k)|^2w_0(x)^2/√(1-x^2)dx,

then the weighted expression

    w_0(x)∏_k=1^n(x-c_k^∗)

is asymptotically alternating on [-1,1], see <cit.>. As it turns out, this information suffices to apply a result due to de la Vallée-Poussin found in <cit.> and ensure that 

    max_x∈ [-1,1]|w_0(x)∏_k=1^n(x-c_k^∗)|∼min_a_1,…,a_nmax_x∈ [-1,1]w_0(x)|∏_k=1^n(x-a_k)|

as n→∞. Note that 

    ∏_k=1^n(x-c_k^∗)
 is nothing but the monic orthogonal polynomial of degree n corresponding to the weight function w_0(x)^2/√(1-x^2) on [-1,1]. By showing that

    max_x∈ [-1,1]|w_0(x)∏_k=1^n(x-c_k^∗)|∼ 2^1-nexp{1/π∫_-1^1log w_0(x)/√(1-x^2)dx}

as n→∞, Bernstein obtains (<ref>) from (<ref>). Consequently, Theorem <ref> is verified in this case. Proceeding from this, Bernstein extends the analysis by allowing for vanishing factors of the form |x-b_k|^s_k to be introduced to the weight where b_k∈ [-1,1] and s_k∈ℝ, see <cit.>. To show this, he uses a clever technique of bounding the factors |x-b_k|^s_k from above and below. However, the connection to the maximal deviation of the corresponding orthogonal polynomials as in (<ref>) gets lost. 

To illustrate why this connection becomes more delicate when zeros are added to the weight function, he provides the explicit example of the monic Jacobi polynomials. Following <cit.>, we let P_n^(α,β) denote the classical Jacobi polynomials with parameters α,β >-1. These are uniquely defined by the property that P_n^(α,β) is a polynomial of exact degree n and

    ∫_-1^1(1-x)^α(1+x)^β P_m^(α,β)(x)P_n^(α,β)(x)dx = 2^α+β+1/2n+α+β+1Γ(n+α+1)Γ(n+β+1)/Γ(n+α+β+1)n!δ_nm,

where δ_nm is the Kronecker delta.

Since all zeros of the Jacobi polynomials reside in [-1,1] and since 

    P_n^(α,β)(x) = 2^-n2n+α+βnx^n+,


we find that

    2^nP_n^(α,β)(x)/2n+α+βn = ∏_k=1^n(x-cosψ_k^∗)

for some values ψ_k^∗∈ [0,π]. Bernstein's analysis provides the following result.

	Let ρ_α = α/2+1/4 and ρ_β = β/2+1/4,  and let ψ_k^∗ be associated with P_n^(α,β) as in (<ref>). 
	
		
  * If 
		
    0≤ρ_α≤ 1/2  and   0≤ρ_β≤ 1/2

		both hold, then
		
    max_x∈ [-1,1](1-x)^ρ_α(1+x)^ρ_β|∏_k=1^n(x-cosψ_k^∗)|∼ 2^1-n-ρ_α-ρ_β

		as n→∞.
		
  * If one of the conditions in (<ref>) fails to hold, then so does (<ref>).
	
	

Note that by replacing w_0 with (1-x)^ρ_α(1+x)^ρ_β in (<ref>), the corresponding weight function, to which the orthogonal polynomials are associated, is nothing but

    [(1-x)^ρ_α(1+x)^ρ_β]^2/√(1-x^2) = (1-x)^α(1+x)^β.

It is a simple matter to verify that

    2^1-nexp{1/π∫_-1^1log [(1-x)^ρ_α(1+x)^ρ_β]/√(1-x^2)dx} = 2^1-n-ρ_α-ρ_β,

see for instance <cit.> where also a translation of the proof of Theorem <ref> appears. As such, we see that the monic Jacobi polynomials defined in (<ref>) are close to minimal with respect to

    min_a_1,…,a_nmax_x∈ [-1,1](1-x)^ρ_α(1+x)^ρ_β|∏_k=1^n(x-a_k)|

precisely when (<ref>) holds. 

As a partial result, we wish to carry out a detailed analysis of the quantity 

    max_x∈ [-1,1](1-x)^ρ_α(1+x)^ρ_β|∏_k=1^n(x-cosψ_k^∗)|

for the parameters satisfying (<ref>). By carefully manipulating a Bernstein-type inequality due to Chow et al. <cit.>, we will establish the following result.

	Let ρ_α = α/2+1/4 and ρ_β = β/2+1/4, and let ψ_k^∗ be as in (<ref>). If (<ref>) holds, then
	
    max_x∈ [-1,1](1-x)^ρ_α(1+x)^ρ_β|∏_k=1^n(x-cosψ_k^∗)|≤ 2^1-n-ρ_α-ρ_β

	for every n with equality if and only if ρ_α,ρ_β∈{0,1/2}.
	

By combining Theorem <ref> with Theorem <ref>, we see that 

    lim_n→∞2^nmax_x∈ [-1,1](1-x)^ρ_α(1+x)^ρ_β|∏_k=1^n(x-cosψ_k^∗)|
      = sup_n2^nmax_x∈ [-1,1](1-x)^ρ_α2(1+x)^ρ_β|∏_k=1^n(x-cosψ_k^∗)|= 2^1-ρ_α-ρ_β

which shows that the convergence is from below.
The question of obtaining upper bounds for the quantity

    max_x∈ [-1,1](1-x)^a(1+x)^b|∏_k=1^nP_n^(α,β)(x)|

for different values of a,b,α,β is well studied, see for instance <cit.> and has applications to e.g. representation theory and the study of Schrödinger operators. Our main interest is the applications that such a result has to the corresponding Chebyshev problem (<ref>). To state our main result, we introduce the Widom factor _n(ρ_α,ρ_β) defined by

    _n(ρ_α,ρ_β):= 2^nmin_a_1,…,a_nmax_x∈ [-1,1](1-x)^ρ_α(1+x)^ρ_β|∏_k=1^n(x-a_k)|.

The choice of naming in honour of H. Widom – who in <cit.> considered Chebyshev polynomials corresponding to a plethora of compact sets in the complex plane – stems from <cit.>. The considerations of Widom factors in the literature are plentiful, see for instance <cit.>. In this article we wish to carry out a detailed study of the behavior of _n(ρ_α,ρ_β) for different parameter values.

	For any value of the parameters ρ_α,ρ_β≥ 0, we have that
		
    _n(ρ_α,ρ_β)∼ 2^1-ρ_α-ρ_β

		as n→∞. Furthermore:
		
			
  * If ρ_α,ρ_β∈{0,1/2}, then the quantity _n(ρ_α,ρ_β) is constant.
			
  * If ρ_α,ρ_β∈ [0,1/2], then
			
    sup_n_n(ρ_α,ρ_β)= 2^1-ρ_α-ρ_β.

			
  * If ρ_α,ρ_β∈{0}∪ [1/2,∞), then
			
    inf_n_n(ρ_α,ρ_β) = 2^1-ρ_α-ρ_β,

			
    sup_n _n(ρ_α,ρ_β)≤(2ρ_α/ρ_α+ρ_β)^ρ_α(2ρ_β/ρ_α+ρ_β)^ρ_β,

			and _n(ρ_α,ρ_β) decreases monotonically as n increases.
		
		

The case of ρ_α,ρ_β∈{0,1/2} is classical and if ρ_β∈{0,1/2} and ρ_α≥ 1/2 (or vice versa), the result was settled in <cit.>. We will in fact apply a similar technique to what is used there for the general setting of ρ_α,ρ_β≥1/2. This method originates with <cit.>. As such, the proofs of the different cases of Theorem <ref> uses essentially different techniques and will be split into three separate sections.



	 We should also note that while negative parameters ρ_α and ρ_β are allowed in Theorem <ref>, they simply result in the fact that a zero of the minimizer is forcedly placed to cancel the effect of the pole and hence the setting transfers to the one of non-negative parameters if the degree of the associated Chebyshev polynomial is sufficiently large.

Asymptotic results for the Chebyshev polynomials corresponding to (<ref>) exist for smooth enough weight functions w which are strictly positive on [-1,1]. See for instance <cit.>. If the weight function w vanishes somewhere on [-1,1], there does not seem to be many general results for the corresponding Chebyshev polynomials apart from Theorem <ref> found in <cit.>.

The extension of Chebyshev polynomials to the complex plane was first carried out by Faber in <cit.>. Apart from the proof of Theorem <ref>, our considerations exclusively concern Chebyshev polynomials corresponding to Jacobi weights and we therefore refrain from a detailed presentation of the general complex case. We note, however, that if ⊂ is a compact set and w:→ [0,∞) is a weight function which is non-zero for at least n points, then there exists a unique monic polynomial T_n^w of degree n minimizing

    max_z∈w(z)|∏_k=1^n(z-a_k)|.

We invite the reader to consult <cit.> for a detailed presentation of complex Chebyshev polynomials.

Our motivation in providing a detailed study of _n(ρ_α,ρ_β) originates from an inquiry of complex Chebyshev polynomials corresponding to star graphs of the form {z:z^m∈ [-2,2]}. It can be shown that these complex Chebyshev polynomials can be directly related to Chebyshev polynomials on [-1,1] with Jacobi weights, see <cit.> for details. In the study of the Widom factors for star graphs, certain monotonicity properties appear which can be explained by Theorem <ref>.  We trust that Theorem <ref> will also be useful in many other connections.




§ THE CASE OF Ρ_Α,Ρ_Β∈{0,1/2}

	We proceed by delving into the proof of Theorem <ref>, split into three cases. The first case is very simple. When ρ_α,ρ_β∈{0,1/2}, the minimizers of (<ref>) are simply the Chebyshev polynomials of the 1^st to 4^th kind, see e.g. <cit.>. Using basic trigonometry, it follows that if θ∈ [0,π] then
		
    cosθ/2    = √(1+cosθ/2), 
    sinθ/2    = √(1-cosθ/2).

		Through the change of variables x = cosθ, the trigonometric functions
		
    T_n(x) = 2^1-ncos nθ,      U_n(x) = 2^-nsin(n+1)θ/sinθ,	
    
    			V_n(x) = 2^-ncos(n+1/2)θ/cosθ/2,       W_n(x) = 2^-nsin(n+1/2)θ/sinθ/2,

		all define monic polynomials in x of degree n. These are precisely, in order, the monic Chebyshev polynomials of the 1^st, 2^nd, 3^rd and 4^th kind. It is a straightforward matter to verify that they all satisfy the alternating property from (<ref>) upon multiplication with the weight functions
		
    1,     sinθ=√(1-x^2), 
    √(2)cosθ/2=√(1+x),     √(2)sinθ/2=√(1-x),

		respectively. As a result, T_n, U_n, V_n, and W_n

are the minimal configurations sought after in (<ref>) for the weight parameters ρ_α,ρ_β∈{0,1/2}. It is evident that _n(ρ_α,ρ_β) = 2^1-ρ_α-ρ_β in all these cases and this shows the first part of Theorem <ref>.
		
	

§ THE CASE OF 0≤Ρ_Α,Ρ_Β≤ 1/2
	 
	Having settled the first case of Theorem <ref>, we turn toward the verification of the second case. For parameter values ρ_α,ρ_β∈ [0,1/2], we want to show that
	
    sup_n_n(ρ_α,ρ_β) = 2^1-ρ_α-ρ_β.

	These parameter values correspond to the lower left square in Figure <ref>. Observe that if ρ_α,ρ_β∈{0,1/2} then this follows from the first case of Theorem <ref> since then the sequence {_n(ρ_α,ρ_β)} is constant. Our approach will consist of first verifying Theorem <ref>. For this reason we remind the reader of our definition of ψ_k^∗∈ [0,π] through the formula 
    ∏_k=1^n(x-cosψ_k^∗)=2^nP_n^(α,β)(x)/2n+α+βn,

	where  P_n^(α,β) denotes the Jacobi polynomial with parameters α,β and the angles ψ_k^∗ are those for which cosψ_1^∗,…,cosψ_n^∗ enumerate its zeros. We also recall the relation between the L^2 and L^∞ parameters
	
    ρ_α=α/2+1/4,
    ρ_β =β/2+1/4.

	If we can show the validity of (<ref>), then we trivially obtain
	
    _n(ρ_α,ρ_β)≤ 2^nmax_x∈ [-1,1](1-x)^ρ_α(1+x)^ρ_β|∏_k=1^n(x-cosψ_k^∗)|≤ 2^1-ρ_α-ρ_β

	from which (<ref>), the second case of Theorem <ref> follows from the fact that _n(ρ_α,ρ_β)∼ 2^1-ρ_α-ρ_β as n→∞. The key to verifying (<ref>) is the following result.
	
		If -1/2≤α,β≤ 1/2 and θ∈[0,π], then 
		
    (sinθ/2)^α+1/2(cosθ/2)^β+1/2|P_n^(α,β)(cosθ)|≤Γ(q+1)/Γ(1/2)n+qn(n+α+β+1/2)^-q-1/2

		where q = max(α,β).
		
	
	This result provides a Bernstein-type inequality for Jacobi polynomials and sharpens a previous result of Baratella <cit.> who had shown (<ref>) with the larger factor 2.821 in place of Γ(q+1)/Γ(1/2). In the case where |α| = |β| = 1/2, the inequality in (<ref>) is saturated. For arbitrary -1/2≤α,β≤ 1/2 the constant Γ(q+1)/Γ(1/2) is asymptotically sharp as can be seen from an argument using Stirling's formula which we will provide in the proof of Lemma <ref>. The type of inequality exemplified in (<ref>) originate with Bernstein who showed in <cit.> that the Legendre polynomials P_n := P_n^(0,0) satisfy
	
    (sinθ)^1/2|P_n(cosθ)|≤(2/π)^1/2n^-1/2,  θ∈ [0,π].

	Antonov and Holšhevnikov <cit.> sharpened (<ref>) and later Lorch <cit.> extended such an inequality to Gegenbauer polynomials. In terms of Jacobi polynomials, this provides a Bernstein-type inequality for P_n^(λ,λ) with -1/2≤λ≤ 1/2. Theorem <ref> contains all these inequalities as special cases. In a different direction, the Erdélyi–Magnus–Nevai Conjecture <cit.> proposes that if P̂_n^(α,β) denotes the orthonormal Jacobi polynomial then for any parameter values α,β≥-1/2,
	
    max_x∈ [-1,1](1-x)^α+1/2(1+x)^β+1/2(P̂_n^(α,β)(x))^2 = O(max[1,(α^2+β^2)^1/4]).

	Theorem <ref> can be used to verify this conjecture in the case where |α|,|β|≤ 1/2. This is shown in <cit.> where an extensive consideration concerning the sharpness of Theorem <ref> is studied numerically for different parameter values.
	
	We consider transferring the trigonometric weights to algebraic weights as in the statement of Theorem <ref>. Letting x=cosθ, we find that
	
    (sinθ/2)^α+1/2   =(1-x/2)^α/2+1/4 = 2^-ρ_α(1-x)^ρ_α,
    (cosθ/2)^β+1/2   =(1+x/2)^β/2+1/4=2^-ρ_β(1+x)^ρ_β,

	with the parameter relation stated in (<ref>).





	Together with (<ref>), (<ref>) and (<ref>), the change of variables transforms (<ref>) into the statement that for x∈ [-1,1],
	
    (1-x)^ρ_α(1+x)^ρ_β|∏_k=1^n(x-cosψ_k^∗)|≤Γ(q+1)2^n+α+β+1/2n+qn/√(π)(n+α+β+1/2)^q+1/22n+α+βn,

	where q = max(α,β). From (<ref>) we see that this implies that 
	
    _n(ρ_α,ρ_β)≤ 2^nΓ(q+1)2^n+α+β+1/2n+qn/√(π)(n+α+β+1/2)^q+1/22n+α+βn

	for ρ_α,ρ_β∈[0,1/2]. In order to determine the upper bound of (<ref>) and (<ref>), we are ultimately led to consider the quantity
	
    M_n(α,β):=Γ(q+1)2^2n+α+β+1/2n+qn/√(π)(n+α+β+1/2)^q+1/22n+α+βn.

	
	
		Suppose that α,β∈ [-1/2,1/2]. Then M_n(α,β), defined in (<ref>), increases monotonically to 2^1-ρ_α-ρ_β as n→∞.
		
	
	The proof we provide constitutes a lengthy computation. We have therefore chosen to illustrate the simpler case where α = β = q in the proof below while postponing most of the remaining computations to the appendix. The asymptotical behavior M_n(α,β)∼ 2^1-ρ_α-ρ_β as n→∞ merits attention since this shows that the constant Γ(q+1)/Γ(1/2) is sharp in (<ref>). To see this, note that
	
    _n(ρ_α,ρ_β)≤ 2^nmax_x∈ [-1,1](1-x)^ρ_α(1+x)^ρ_β|∏_k=1^n(x-cosψ_k^∗)|≤ M_n(α,β).

	The left-hand side converges to 2^1-ρ_α-ρ_β as n→∞ and consequently
				
    lim inf_n→∞M_n(α,β)≥ 2^1-ρ_α-ρ_β.

				 On the other hand, as Lemma <ref> shows, M_n(α,β)∼ 2^1-ρ_α-ρ_β as n→∞ and therefore the constant factor in (<ref>) cannot be improved.
	
			
				We rewrite the binomial coefficients using Gamma functions so that
				
    n+qn=Γ(n+q+1)/Γ(n+1)Γ(q+1),

				
    2n+α+βn=Γ(2n+α+β+1)/Γ(n+1)Γ(n+α+β+1).

				The Legendre duplication formula implies that
				
    Γ(2n+α+β+1)=2^2n+α+β/√(π)Γ(n+α+β+1/2)Γ(n+α+β/2+1).

				The combination of (<ref>), (<ref>), (<ref>) and (<ref>) yields
				
    M_n(α,β)   =2^2n+α+β+1/2Γ(n+q+1)Γ(n+α+β+1)/√(π)(n+α+β+1/2)^q+1/2Γ(2n+α+β+1)
       =2^1-α-β/2Γ(n+q+1)Γ(n+α+β+1)/(n+α+β+1/2)^q+1/2Γ(n+α+β+1/2)Γ(n+α+β/2+1).

				The Gamma function satisfies that Γ(n+a)/Γ(n+b) ∼ n^a-b as n→∞, and we may conclude that
				
    M_n(α,β)∼2^1-α-β/2n^q+1n^α+β+1/n^q+1/2n^α+β+1/2n^α+β/2+1=2^1-α-β/2=2^1-ρ_α-ρ_β

				as n→∞. 				
				  
				We are left to determine the monotonicity of M_n(α,β). For this reason we consider the quotient 
				
    M_n+1(α,β)/M_n(α,β)   =(n+q+1)(n+α+β+1)(n+α+β+1/2)^q+1/2/(n+α+β+1/2+1)^q+1/2(n+α+β+1/2)(n+α+β/2+1)
       =(n+q+1)(n+α+β+1)(n+α+β+1/2)^q-1/2/(n+α+β+1/2+1)^q+1/2(n+α+β/2+1).

				Whether this quotient is bigger than or smaller than 1 determines the monotonicity of the sequence {M_n(α,β)}. To simplify, we begin by considering the case α=β = q. Writing M_n(q):=M_n(q,q), the quotient (<ref>) becomes
				
    M_n+1(q)/M_n(q)    = (n+1+q)(n+2q+1)/(n+q+1/2)(n+q+1)(n+q+1/2/n+q+3/2)^q+1/2
        = (n+2q+1)(n+q+1/2)^q-1/2/(n+q+3/2)^q+1/2.

				We introduce the function
				
    f(x) = (x+2q+1)(x+q+1/2)^q-1/2/(x+q+3/2)^q+1/2,

				and claim that f'(x)<0 for x>0. From the fact that f(x)→ 1 as x→∞, this will imply that f(x)≥ 1 for every x>0. As a consequence we see that M_n+1(q)≥ f(n)M_n(q)≥ M_n(q).
				
				Taking the logarithmic derivative of f generates
				
    f'(x)/f(x) = q^2-1/4/(x+2q+1)(x+q+1/2)(x+q+3/2).








				If -1/2<q<1/2, then it is clear that 
				
    f'(x)/f(x)<0

				and as a consequence f'(x)<0. We conclude that M_n+1(q)≥ M_n(q) and this settles the case where α= β = q. 
				
				If α≠β, then the problem is more delicate. We again need to show that the quotient (<ref>) is strictly greater than 1. In order to show this, we reintroduce the function f by defining it as
			
    f(x)=(x+q+1)(x+α+β+1)/(x+α+β/2+1)(x+α+β+1/2)^q-1/2/(x+α+β+1/2+1)^q+1/2.

			In a completely analogous manner, we only need to show that f'(x)<0 for every x∈ (0,∞) in order to conclude that f(x)≥ 1 and consequently M_n+1(α,β)≥ f(n)M_n(α,β)≥ M_n(α,β). Since M_n is symmetric with respect to its variables and q = max(α,β), it is enough to consider the parameter values -1/2≤β≤α≤ 1/2.
			 
			The logarithmic derivative of f is given by
			
    f'(x)/f(x)=1/x+q+1+1/x+α+β+1+q-1/2/x+α+β+1/2-1/x+α+β/2+1-q+1/2/x+α+β+1/2+1.

			Written over common denominator and utilizing that α = q, this expression becomes
			
    f'(x)/f(x)=c_2(α,β)x^2+c_1(α,β)x+c_0(α,β)/(x+α+1)(x+α+β+1)(x+α+β/2)(x+α+β/2+1)(x+α+β/2+3/2),

			where
			
    c_2(α,β)   =α^2/2+β^2/2-1/4,
    
    			c_1(α,β)   =3α^3/4+(4β+8)α^2/8+(β^2-1)α/4+β^3/2+β^2-β/4-1/2,
    
    			c_0(α,β)   =α^4/4+(3β+6)α^3/8+(β+2)^2α^2/8+(β^2-1)(β+2)α/8+β^4/8+β^3/2+3β^2/8-β/4-1/4.

			Now it is immediately clear that c_2(α,β)<0 unless |α|=|β|=1/2. We claim that the same is true for the remaining coefficients c_1(α,β) and c_0(α,β). Namely, c_k(α,β)<0 for k=0,1,2 unless |α|=|β| = 1/2. A consequence of these inequalities is that f'(x)<0 for x>0 which is precisely the sought after inequality. The proof of this is provided in Lemma <ref> in the appendix. As a result, we obtain that f(n)≥ 1 and therefore
			
    M_n+1(α,β)=f(n)M_n(α,β)≥ M_n(α,β).

			In conclusion, M_n(α,β) converges monotonically to 2^1-ρ_α-ρ_β from below. 
			
			An immediate consequence of Lemma <ref> together with (<ref>) is that 
			
    _n(ρ_α,ρ_β)≤ 2^nmax_x∈ [-1,1](1-x)^ρ_α(1+x)^ρ_β|∏_k=1^n(x-cosψ_k^∗)|≤ M_n(α,β)≤ 2^1-ρ_α-ρ_β.

			This simultaneously shows that Theorem <ref> holds and that 
			
    _n(ρ_α,ρ_β)≤ 2^1-ρ_α-ρ_β

			is valid for every n. On the other hand, (<ref>) implies that these quantities are all asymptotically equal as n→∞. This is possible only if
			
    sup_n_n(ρ_α,ρ_β)= 2^1-ρ_α-ρ_β

			and hence the second case of Theorem <ref> is shown. From the fact that _n(ρ_α,ρ_β)≤ M_n(α,β), we gather the following corollary to Lemma <ref>.
			
				If 0≤ρ_α, ρ_β≤ 1/2 and r=2max(ρ_α, ρ_β), then
				
    _n(ρ_α,ρ_β)≤Γ(r+1/2)2^2n+ρ_α+ρ_βn+r-1/2n/√(π)(n+ρ_α+ρ_β)^r2n+2ρ_α+2ρ_β-1n≤ 2^1-ρ_α-ρ_β

				and the rightmost inequality is strict unless ρ_α, ρ_β∈{0,1/2}.
			



§ THE CASE OF Ρ_Α,Ρ_Β≥ 1/2

	
	We are left to prove the third and final case of Theorem <ref>. Our approach to proving this revolves around relating Jacobi weighted Chebyshev polynomials to weighted Chebyshev polynomials on the unit circle. These ideas have previously been investigated in <cit.> for certain choices of the parameters. We will provide a general argument which works for any choice of parameters ρ_α,ρ_β≥ 1/2. 
	
	The approach first studied in <cit.> hinges upon the Erdős–Lax inequality, see <cit.>. If P is a polynomial of degree n which does not vanish anywhere on , then
	
    max_|z| = 1|P'(z)| ≤n/2max_|z| = 1|P(z)|.

	This result relates the maximum modulus of a polynomial with its derivative. Furthermore, equality occurs in (<ref>) if and only if all zeros of P lie on the unit circle. A recent extension of (<ref>) to the case of generalized polynomials having all zeros situated on the unit circle is the following.
	
	
		Let s_k≥ 1 and θ_k∈ [0,2π) for k = 1,…,n. Then
		
    max_|z| = 1|d/dz{∏_k=1^n(z-e^iθ_k)^s_k}| = ∑_k=1^ns_k/2max_|z| = 1|∏_k=1^n(z-e^iθ_k)^s_k|.

	
	Given a polynomial with zeros in the unit disk, there is a procedure of constructing a related polynomial whose zeros all lie on the unit circle.
	
		Let |a_k|≤ 1 for k = 1,…,n and suppose that m∈ℤ_+ and θ∈. Then
		
    z^m∏_k=1^n(z-a_k)+e^iθ∏_k=1^n(1-a_k z)

		does not vanish away from |z| = 1.
		
	
	
	Utilizing these two results, we obtain the following theorem which “opens up” the interval to the circle.
	
	
		Given ρ_α,ρ_β≥ 1/2, let θ_1^∗,…,θ_n^∗∈ [0,2π) be the unique minimizing data of the expression
		
    I_n:=min_θ_1,…,θ_nmax_x∈ [-1,1]|(1-x)^ρ_α(1+x)^ρ_β∏_k=1^n(x-cosθ_k)|.

		In that case, 
		
    1/2ρ_α+2ρ_β+2nd/dz{(z-1)^2ρ_α(z+1)^2ρ_β∏_k=1^n(z-e^iθ_k^∗)(z-e^-iθ_k^∗)}

		is the unique minimizer of 
		
    C_n:=min_a_1,…,a_2n+1max_|z| = 1|(z-1)^2ρ_α-1(z+1)^2ρ_β-1∏_k=1^2n+1(z-a_k)|.

		
		Furthermore, we have that
		
    C_n = 2^n+ρ_α+ρ_β-1 I_n.





	
	
	
		We will prove this result in the opposite direction as it is stated by first analyzing the minimizer of (<ref>). Assume that the points a_k^∗ are uniquely chosen to satisfy
		
    max_|z| = 1|(z-1)^2ρ_α-1(z+1)^2ρ_β-1∏_k=1^2n+1(z-a_k^∗)| = C_n.

		A result of Fejér <cit.> says that the zeros of a complex Chebyshev polynomial corresponding to a compact set  are always situated in the convex hull of . In our setting, a similar reasoning ensures us that |a_k^∗|≤ 1 holds for k = 1,… 2n+1. It is actually possible to show that this inequality is strict, see <cit.> for details. We form the combination
		
    P(z) = z∏_k=1^2n+1(z-a_k^∗)-∏_k=1^2n+1(1-a_k^∗ z).

		It is immediate that P is a monic polynomial of degree 2n+2 and by Lemma <ref> we conclude that P has all its zeros on the unit circle. Since the minimizer of (<ref>) is unique, all a_k^∗ must come in conjugate pairs with one possible exception. Due to symmetry of the weight function, this exceptional zero must always be real.
		
		Therefore,
		
    P(z) = z∏_k=1^2n+1(z-a_k^∗)-∏_k=1^2n+1(1-a_k^∗ z)

		and as a consequence,
		
    P(1)     = ∏_k=1^2n+1(1-a_k^∗)-∏_k=1^2n+1(1-a_k^∗) = 0, 
    
    			P(-1)     = -∏_k=1^2n+1(-1-a_k^∗)-∏_k=1^2n+1(1+a_k^∗) = 0.

		We conclude that
		
    P(z) = (z-1)(z+1)∏_k=1^2n(z-e^iθ_k^∗)

		for some values θ_k^∗∈ [0,2π). 
		
		Now let φ_k∈ [0,2π) be chosen such that
		
    max_|z| = 1|(z-1)^2ρ_α(z+1)^2ρ_β∏_k=1^2n(z-e^iφ_k)| = min_θ_1,…,θ_2nmax_|z| = 1|(z-1)^2ρ_α(z+1)^2ρ_β∏_k=1^2n(z-e^iθ_k)|.

		From the fact that 
		
    ∏_k=1^2n+1|1-a_k^∗ z| = ∏_k=1^2n+1|z-a_k^∗|

		for |z| = 1 together with the representation in (<ref>), we conclude from the triangle inequality that
		
    max_|z| = 1|(z-1)^2ρ_α(z+1)^2ρ_β∏_k=1^2n(z-e^iφ_k)| ≤max_|z| = 1|(z-1)^2ρ_α-1(z+1)^2ρ_β-1P(z)|
    ≤ 2 max_|z| = 1|(z-1)^2ρ_α-1(z+1)^2ρ_β-1∏_k=1^2n+1(z-a_k^∗)| = 2 C_n.

		On the other hand, the expression
		
    1/2ρ_α+2ρ_β+2nd/dz(z-1)^2ρ_α(z+1)^2ρ_β∏_k=1^2n(z-e^iφ_k)

		is of the form
		
    (z-1)^2ρ_α-1(z+1)^2ρ_β-1∏_k=1^2n+1(z-b_k)

		for some values of b_k∈. From the Gauss–Lucas Theorem we can actually conclude that |b_k|≤ 1. As it turns out, this derivative is a candidate for a minimizer of (<ref>). Therefore,
		
    max_|z|=1|1/2ρ_α+2ρ_β+2nd/dz{(z-1)^2ρ_α(z+1)^2ρ_β∏_k=1^2n(z-e^iφ_k)}|
    ≥max_|z| = 1|(z-1)^2ρ_α-1(z+1)^2ρ_β-1∏_k=1^2n+1(z-a_k^∗)|

		and Theorem <ref> immediately gives us that 
		
    2max_|z|=1|1/2ρ_α+2ρ_β+2nd/dz{(z-1)^2ρ_α(z+1)^2ρ_β∏_k=1^2n(z-e^iφ_k)}|
     = max_|z| = 1|(z-1)^2ρ_α(z+1)^2ρ_β∏_k=1^2n(z-e^iφ_k)|.

		By combining (<ref>), (<ref>) and (<ref>), we conclude that equality must hold in all the inequalities. 
		In particular,
		
    max_|z|=1|1/2ρ_α+2ρ_β+2nd/dz{(z-1)^2ρ_α(z+1)^2ρ_β∏_k=1^2n(z-e^iφ_k)}|
     = max_|z| = 1|(z-1)^2ρ_α-1(z+1)^2ρ_β-1∏_k=1^2n+1(z-a_k^∗)|

		and 
		
    max_|z| = 1|(z-1)^2ρ_α(z+1)^2ρ_β∏_k=1^2n(z-e^iφ_k)| = max_|z| = 1|(z-1)^2ρ_α-1(z+1)^2ρ_β-1P(z)|.

		As a consequence of (<ref>) together with uniqueness of the minimizer, we have that 
		
    1/2ρ_α+2ρ_β+2nd/dz{(z-1)^2ρ_α(z+1)^2ρ_β∏_k=1^2n(z-e^iφ_k)} =(z-1)^2ρ_α-1(z+1)^2ρ_β-1∏_k=1^2n+1(z-a_k^∗)

		and so the minimizing points φ_k, which were chosen to satisfy (<ref>), are determined from the point set {a_k^∗}. Incidentally we have established that the solution to (<ref>) is unique and as a consequence all nodes e^iφ_k come in conjugate pairs. Furthermore, (<ref>) implies that after a possible rearrangement, it holds that θ_k^∗ = φ_k for k = 1,…,2n.
		
		We now let x= (z+z^-1)/2 and recall that x∈ [-1,1] when |z| = 1. Under this transformation, 
		
    (z-e^iθ_k)(z-e^-iθ_k) = 2z(x-cosθ_k)

		so if we arrange the angles θ_k^∗ in such a way that θ_k+n^∗ = θ_k^∗+π for k=1,…,n, then
		
    (z-1)^2ρ_α(z+1)^2ρ_β∏_k=1^2n(z-e^iθ_k∗) = (2z)^ρ_α+ρ_β+n(x-1)^ρ_α(x+1)^ρ_β∏_k=1^n(x-cosθ_k^∗).

		In conclusion, to each expression of the form
		
    (x-1)^ρ_α(x+1)^ρ_β∏_k=1^n(x-cosθ_k),

		there corresponds a function
		
    (z-1)^2ρ_α(z+1)^2ρ_β∏_k=1^n(z-e^iθ_k)(z-e^-iθ_k)

		and their absolute values are related through multiplication with the constant 2^ρ_α+ρ_β+n. Since the left-hand side of (<ref>) is minimal in terms of maximal modulus on the unit circle, we conclude that 
		
    max_x∈ [-1,1]|(x-1)^ρ_α(x+1)^ρ_β∏_k=1^n(x-cosθ_k^∗)| = I_n.

		The result now follows from the uniqueness of the minimizer of (<ref>).
	
	We end this section by listing some consequences of Theorem <ref>.
	
		When ρ_α,ρ_β≥ 1/2, there is a unique configuration  θ_1^∗,…,θ_2n^∗∈ [0,2π) such that 
		
    max_|z| = 1|(z-1)^2ρ_α(z+1)^2ρ_β∏_k=1^2n(z-e^iθ_k^∗)| = min_θ_1,…,θ_2nmax_|z| = 1|(z-1)^2ρ_α(z+1)^2ρ_β∏_k=1^2n(z-e^iθ_k)|.

	
	Our main conclusion is the following monotonicity result which constitutes the final puzzle piece to proving the third case in Theorem <ref>.
	
		If ρ_α,ρ_β≥ 1/2, then _n(ρ_α,ρ_β) decays monotonically to 2^1-ρ_α-ρ_β as n→∞ and for every n, we have that
		
    _n(ρ_α,ρ_β)≤(2ρ_α/ρ_α+ρ_β)^ρ_α(2ρ_β/ρ_α+ρ_β)^ρ_β.

	
	
		We begin by verifying the monotonicity of the sequence of Widom factors. From (<ref>) and Theorem <ref> we know that
		
    _n(ρ_α,ρ_β) = 
    		2^nmin_θ_1,…,θ_nmax_x∈ [-1,1](1-x)^ρ_α(1+x)^ρ_β|∏_k=1^n(x-cosθ_k)| 
     =
    		2^1-ρ_α-ρ_βmin_a_1,…,a_2n+1max_|z| = 1|(z-1)^2ρ_α-1(z+1)^2ρ_β-1∏_k=1^2n+1(z-a_k)|.

Clearly the right-hand side decreases monotonically in n since multiplying by z^2 does not change the absolute value and
		
    max_|z|=1|z^2(z-1)^2ρ_α-1(z+1)^2ρ_β-1∏_k=1^2n+1(z-a_k^∗)| 
    ≥min_a_1,…,a_2(n+1)+1max_|z| = 1|(z-1)^2ρ_α-1(z+1)^2ρ_β-1∏_k=1^2(n+1)+1(z-a_k)|.

Hence monotonicity of _n(ρ_α,ρ_β) follows.
		
		To show the upper bound, we note that Theorem <ref> is still valid if n = 0 meaning that the expression
	
    1/2ρ_α+2ρ_βd/dz{(z-1)^2ρ_α(z+1)^2ρ_β} = (z-1)^2ρ_α-1(z+1)^2ρ_β-1(z-a^∗)

	minimizes (<ref>) in the case where n=0. Consequently,
	
    2^1-ρ_α-ρ_βmin_amax_|z|=1|(z-1)^2ρ_α-1(z+1)^2ρ_β-1(z-a)|  = max_x∈ [-1,1]|(1-x)^ρ_α(1+x)^ρ_β| 
    	
    	 = (1+ρ_α-ρ_β/ρ_α+ρ_β)^ρ_α(1-ρ_α-ρ_β/ρ_α+ρ_β)^ρ_β = (2ρ_α/ρ_α+ρ_β)^ρ_α(2ρ_β/ρ_α+ρ_β)^ρ_β

	and the upper bound now follows from the monotonicity of _n(ρ_α,ρ_β). 
	
		

§ WHAT ABOUT THE REMAINING PARAMETERS Ρ_Α,Ρ_Β?

	To conclude our investigation we question what can be said concerning the monotonicity properties of _n(ρ_α,ρ_β) for the remaining values of the parameters.  These are precisely those which satisfy 
	
    0<ρ_α<1/2   and   ρ_β≥ 1/2  or   0<ρ_β<1/2   and   ρ_α≥ 1/2,

	as illustrated by the blank strips in Figure <ref>.
	
	


	First of all, we note the following continuity of the Widom factors. We mention in passing that a general result for the continuity of L^p Widom factors with p<∞ is provided in <cit.>.
	
		
		For fixed n≥ 0, the Widom factor _n(ρ_α,ρ_β) is continuous in ρ_α,ρ_β≥ 0.
	
	
		Let θ̃_k^∗ and θ_k^∗ denote the minimizing nodes associated with the parameters (ρ̃_α,ρ̃_β) and (ρ_α,ρ_β), respectively. For any choice of parameters we have
		
    max_x∈[-1,1]|(1-x)^ρ̃_α(1+x)^ρ̃_β∏_k=1^n(x-cosθ̃_k^∗)|   ≤max_x∈[-1,1]|(1-x)^ρ̃_α(1+x)^ρ̃_β∏_k=1^n(x-cosθ_k^∗)|

		and vice versa when the roles 
		of the parameters (ρ_α,ρ_β) and (ρ̃_α,ρ̃_β) are interchanged.
		
		Given ε>0 it is always possible to choose (ρ̃_α, ρ̃_β) close enough to (ρ_α,ρ_β) in a manner that, independently of the choice of a_k∈ [-1,1], it holds that
		
    |max_x∈[-1,1]|(1-x)^ρ_α(1+x)^ρ_β∏_k=1^n(x-a_k)|-max_x∈[-1,1]|(1-x)^ρ̃_α(1+x)^ρ̃_β∏_k=1^n(x-a_k)||≤ε.

			We thus find that 
	
    max_x∈[-1,1]|(1-x)^ρ̃_α(1+x)^ρ̃_β∏_k=1^n(x-cosθ̃_k^∗)|   ≤max_x∈[-1,1]|(1-x)^ρ̃_α(1+x)^ρ̃_β∏_k=1^n(x-cosθ_k^∗)| 
       ≤max_x∈[-1,1]|(1-x)^ρ_α(1+x)^ρ_β∏_k=1^n(x-cosθ_k^∗)|+ε

		and therefore
		
    max_x∈[-1,1]|(1-x)^ρ̃_α(1+x)^ρ̃_β∏_k=1^n(x-cosθ̃_k^∗)|-max_x∈[-1,1]|(1-x)^ρ_α(1+x)^ρ_β∏_k=1^n(x-cosθ_k^∗)|≤ε.

		By symmetry, we may also deduce that the negative of the left-hand side is ≤ε

		when (ρ̃_α, ρ̃_β) is sufficiently close to (ρ_α,ρ_β). This implies the desired continuity of _n(ρ_α,ρ_β) with respect to the parameters ρ_α,ρ_β. 
		
		

	When ρ_α,ρ_β∈ [0,1/2], we know from Theorem <ref> (part 2) that 
    _n(ρ_α,ρ_β)≤ 2^1-ρ_α-ρ_β.
 
	
The same theorem (part 3) shows that	
	
    _n(ρ_α,ρ_β)≥ 2^1-ρ_α-ρ_β

	for ρ_α,ρ_β≥ 1/2. In particular, for a fixed n, Proposition <ref> implies the existence of parameters where the equality 
    _n(ρ_α,ρ_β) = 2^1-ρ_α-ρ_β
 is attained. In fact, such parameters exist on any arc (in the first quadrant) connecting a point of [0,1/2]× [0,1/2] with a point in [1/2,∞)×[1/2,∞). However, these parameters may very well be n dependent. A natural question is for which parameters such an equality occurs.
	
	If we allow ourselves to assume that (<ref>) can be extended slightly outside the parameter domain (ρ_α, ρ_β)∈ [0, 1/2] × [0, 1/2], it is natural to consider parameters for which (<ref>) vanishes. This occurs precisely on the circular segment 
	
    {(ρ_α,ρ_β) = (1/4+cosθ/√(8),1/4 +sinθ/√(8)): θ∈[-π/4,3π/4] }

	which is illustrated in Figure <ref>. To be specific, we recall that M_n(α,β) defined in (<ref>) provides an upper bound of _n(ρ_α,ρ_β) and that
	
    M_n+1(α,β)/M_n(α,β) = (n+q+1)(n+α+β+1)(n+α+β+1/2)^q-1/2/(n+α+β+1/2+1)^q+1/2(n+α+β/2+1).

	By replacing all occurrences of the natural number n with the real variable x on the right-hand side, we obtain the function f(x) from (<ref>). We previously saw in (<ref>) that 
	
    f'(x)/f(x)=c_2(α,β)x^2+c_1(α,β)x+c_0(α,β)/(x+α+1)(x+α+β+1)(x+α+β/2)(x+α+β/2+1)(x+α+β/2+3/2)

	and that the leading term c_2(α,β)x^2 vanishes precisely when |α| = |β| = 1/2. This indicates that the convergence of f'(x)/f(x)→ 0 as x→∞ is “rapid” in this case. If α^2+β^2>1/4, then f'(x)/f(x) will eventually be positive if x is large enough. We therefore believe that the monotonicity properties of M_n(α,β) change somewhere in the vicinity of the circle 
	
    {(α,β):α,β≥ 0, α^2+β^2 = 1/4}.

	Replacing the variables (α,β) by (ρ_α,ρ_β) transfers the circular segment (<ref>) precisely to (<ref>). 
	

	Since we do not know how far the inequality
	
    _n(ρ_α,ρ_β)≤ M_n(α,β)

	can be extended outside of -1/2≤α,β≤ 1/2, our theoretical approach cannot be applied to study this case. We resort to a numerical investigation of the behavior of the corresponding Widom factors using a generalization of the Remez algorithm due to Tang <cit.>. See also <cit.> for an overview on how this algorithm can be applied to the numerical study of Chebyshev polynomials. The numerical results are illustrated in Figure <ref>. It is clearly suggested by these plots that in the vicinity of the set specified in (<ref>), the monotonicity properties of _n(ρ_α,ρ_β) seem to shift from monotonically increasing to monotonically decreasing as we move from the inside of the disk to its exterior. Since these considerations are based on numerical experiments, we are left to speculate if this is indeed the true picture. We formulate the following conjecture.
	
	
	
		Suppose that ρ_α, ρ_β≥ 0.
		
			
  * If 
		
    (ρ_α-1/4)^2+(ρ_β-1/4)^2< 1/8,

		then _n(ρ_α,ρ_β) is monotonically increasing to 2^1-ρ_α-ρ_β as n→∞.
		
  * If
		
    (ρ_α-1/4)^2+(ρ_β-1/4)^2> 1.184/8,

		then _n(ρ_α,ρ_β) is monotonically decreasing to 2^1-ρ_α-ρ_β as n→∞.
		
	

	
	
	
	

§ APPENDIX

	For the benefit of the reader we restate the definitions of the terms c_0(α,β) and c_1(α,β) as defined in (<ref>) and (<ref>).
	
	
    c_1(α,β)   =3α^3/4+(4β+8)α^2/8+(β^2-1)α/4+β^3/2+β^2-β/4-1/2,
    
    			c_0(α,β)   =α^4/4+(3β+6)α^3/8+(β+2)^2α^2/8+(β^2-1)(β+2)α/8+β^4/8+β^3/2+3β^2/8-β/4-1/4.

	We aim to show that these are upper bounded by 0 in the parameter domain {-1/2≤β≤α≤ 1/2}. This is a necessary part of proving Lemma <ref>.
	
		If -1/2≤β≤α≤ 1/2, then c_k(α,β)≤ 0 for k = 1,2 with equality if and only if |α| = |β| = 1/2.
		
	
	
		To show that this is indeed true, we perform a detailed analysis of the coefficients c_0(α,β),  c_1(α,β) on the set {-1/2≤β≤α≤1/2} illustrated in Figure <ref>.

The boundary of {(α,β):-1/2≤β≤α≤ 1/2} is parametrized with t∈ [0,1] through			
			
    c_0(1/2,t-1/2)   =1/8(t+5/2),(t+1)t(t-1),
    
    				c_0(t-1/2,-1/2)   =t(t-1)(1/4t^2+5/16t+1/8),
    
    				c_0(t-1/2,t-1/2)   =(t+1/2)^2t(t-1),
    
    				c_1(1/2,t-1/2)   =t(t-1)(t/2+7/8),
    
    				c_1(t-1/2,-1/2)   =3/4t(t-1)(t+1/2),
    
    				c_1(t-1/2,t-1/2)   =2(t+1/2)t(t-1).

		
			Based on these factorizations it becomes apparent that c_0<0 and c_1<0 holds for t∈ (0,1) but these values of t are precisely those which correspond to the boundary of {(α,β):-1/2≤β≤α≤ 1/2} with the vertices removed. 
			
			Differentiating c_k(α,β) for k = 0,1 with respect to α yields
			
    ∂/∂αc_0(α,β)   = α^3+9(β+2)/8α^2+(β+2)^2/4α-(1-β^2)(β+2)/8,
    ∂^2/∂α^2c_0(α,β)     = 3α^2+9/4(β+2)α+(β+2)^2/4,
    ∂^3/∂α^3c_0(α,β)     = 6α+9/4(β+2),
    ∂/∂αc_1(α,β)   = 9/4α^2+(β+2)α-1-β^2/4.

			
			We begin with showing that c_0(α,β)<0 unless the parameters belong to the vertices of {(α,β):-1/2≤β≤α≤ 1/2}. Assuming that α,β∈ [-1/2,1/2] we obtain from (<ref>) that
			
    ∂^3/∂α^3c_0(α,β)=6α+9/4(β+2)≥-3-9/8+9/2=3/8.

			In particular, this implies that
			
    α↦∂/∂αc_0(α,β)

			is a convex function on [-1/2,1/2] for any fixed β∈ [-1/2,1/2]. Consequently, if β≤ 0 then
			
    max_α∈ [β,0]∂/∂αc_0(α,β)≤max{∂/∂αc_0(β,β),-(1-β^2)(β+2)/8}.

			Clearly, the quantity
			
    -(1-β^2)(β+2)/8

			is negative if -1/2≤β≤ 1/2. We claim that the same is true for ∂/∂αc_0(β,β). To see this we introduce the function h(β) = ∂/∂αc_0(β,β) which is given by
			
    h(β)    = β^3+9/8(β+2)β^2+(β+2)^2/4β-(1-β^2)(β+2)/8 .

			Differentiating two times we obtain
			
    h”(β) = 15β+7.

			It is evident that h(β) is convex if β≥-7/15. On the other hand, if -1/2≤β≤ -7/15 then
			
    h'(β) = 15/2β^2+7β+7/8≤15/8-49/15+7/8<0

			which shows that h is decreasing on [-1/2,-7/15]. Consequently, if -1/2≤β≤ 0 we have
			
    ∂/∂αc_0(β,β) = h(β)≤max{h(-1/2),h(0)} =max{-1/8,-(1-β^2)(β+2)/8}<0.

			By refering to (<ref>) we obtain
			
    max_α∈ [β,0]∂/∂αc_0(α,β)<0

			and therefore c_0(α,β) is strictly decreasing on [β,0]. From (<ref>) we gather that if β∈ [-1/2,1/2] then for any α≥0,
			
    ∂^2/∂α^2c_0(α,β) = 3α^2+9/4(β+2)α+(β+2)^2/4≥ 0.

			This implies that α↦ c_0(α,β) defines a convex function on [0,1/2] and since we already concluded that α↦ c_0(α,β) is strictly decreasing on [β,0], we find that for any fixed β∈ [-1/2,1/2]
			
    max_α∈ [β,1/2]c_0(α,β) = max{c_0(β,β),c_0(1/2,β)}≤ 0

			and c_0(α,β) =0 if and only if α,β∈{-1/2,1/2}. This in particular shows that c_0(α,β)≤ 0 with equality if and only if α = β = ± 1/2 or α = 1/2 and β = -1/2.
			
			We now proceed to study the coefficient c_1(α,β) and show that c_1(α,β)≤ 0 holds on the parameter domain {(α,β):-1/2≤β≤α≤ 1/2}. This turns out to be straightforward. From (<ref>) we find that
			
    ∂/∂αc_1(α,β)  = 9/4α^2+(4β+8)/4α+(β^2-1)/4
        = 9/4(α+2β+4/9+√((2β+4/9)^2+1-β^2/9))(α-2β+4/9+√((2β+4/9)^2+1-β^2/9)).

			Since -1/2≤β≤α≤ 1/2, we have
			
    (α+2β+4/9+√((2β+4/9)^2+1-β^2/9))≥ -1/2+2β+4/9+2β+4/9≥ -1/2+2·3/9 = 2/3-1/2>0

			and consequently the sign of ∂/∂αc_1(α,β) coincides with that of
			
    (α-2β+4/9+√((2β+4/9)^2+1-β^2/9)).

			This is sufficient to conclude that the maximum of c_1(α,β) on [β,1/2] must be attained at an endpoint of [β,1/2] and we gather that
			
    max_α∈ [β,1/2]c_1(α,β) = max{c_1(β,β),c_1(1/2,β)}≤ 0

			with equality attained if and only if α = β = ± 1/2.









§ ACKNOWLEDGEMENT

This study originated from discussions held during a SQuaRE meeting at the American Institute of Mathematics (AIM) facility in Pasadena, CA. We extend our heartfelt gratitude to AIM for their generous hospitality and for creating such an inspiring scientific environment.


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