Complete monotonicity and zeros of sums of squared Baskakov functions

Abstract

Keywords:Baskakov operator Complete monotonicity Convexity Chebyshev-Grüss-type inequality Distribution of zeros Complete elliptic integral of the first kind a b s t r a c tWe prove complete monotonicity of sums of squares of generalized Baskakov basis functions by deriving the corresponding results for hypergeometric functions. Moreover, in the central Baskakov case we study the distribution of the complex zeros for large values of a parameter. We finally discuss the extension of some results for sums of higher powers.

Entropies and Heun functions associated with positive linear operators

“…For c ≥ 0 this conjecture has been validated by Abel et al [1]. In fact, they proved a stronger result:…”

“…Conjecture 6 (in Section 3) was only partially validated in [1]. Other conjectures formulated in [22] wait for solutions.…”

“…The analog of S(x) for positive linear integral operators is discussed in [22]. The results presented in [1] can be used in order to study the Rényi and Tsallis entropies of arbitrary order. The parameters of the Heun function from (16) satisfy the condition q = aαβ; consequently, the properties of this function derived from Case 3 in [14] can be studied.…”

“…The functions p 1] are the fundamental Bernstein polynomials, or shortly the Bernstein basis; see [5, 5.2.5]. For the sake of simplicity we shall use the notation …”

Complete monotonicity of some entropies

“…A stronger conjecture was formulated in [14] and [17]: It was validated for c ≥ 0 by U. Abel, W. Gawronski and Th. Neuschel [1], who proved a stronger result:…”