Abstract. Lommel's function s μ,ν (z) is a particular solution of the differential equation z 2 y + zy + (z 2 − ν 2 )y = z μ+1 . Here we present estimates and monotonicity properties of the positive zeros of s μ−1/2,1/2 (z) when μ ∈ (0, 1). The positivity of a closely related integral is also considered.
We completely determine the set of (α, β) ∈ R 2 for which the function e αx −e βx e x −1 is convex on (0, ∞) and use this result to give some special classes of completely monotonic functions of positive order related to gamma and psi functions.
We completely determine the set of s, t > 0 for which the function L s,t (x) := x− Γ(x+t) Γ(x+s) x s−t+1 is a Bernstein function, that is L s,t (x) is positive with completely monotonic derivative on (0, ∞). The complete monotonicity of several closely related functions is also established.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1098 STAMATIS KOUMANDOS AND MARTIN LAMPRECHT Some important subclasses of completely monotonic functions have been considered in [14]. Koumandos and Pedersen [14] called a function f : (0, ∞) → R completely monotonic of order α if x α f (x) is completely monotonic on (0, ∞).We recall also that the Riemann-Liouville fractional integral I α (m)(t) of order α > 0, of a Borel measure m on [0, ∞) is defined byThe following characterization of completely monotonic functions of positive order was established in [14, Theorem 1.3]. Theorem 1.1. The function f : (0, ∞) → R is completely monotonic of order α > 0 if and only if f is the Laplace transform of a fractional integral of order α of a positive Radon measure m on [0, ∞), that is,The following special case will be used in the sequel. Corollary 1.2. Let r be an integer ≥ 2. The function f (x) is completely monotonic of order r on (0, ∞) if and only if f (x) = ∞ 0 e −xt p(t) dt , where the integral converges for all x > 0 and p(t) is r − 2 times continuously differentiable on [0, ∞) with p (r−2) (t) = t 0
We present a Suffridge-like extension of the Grace-Szegö convolution theorem for polynomials and entire functions with only real zeros. Our results can also be seen as a q-extension of Pólya's and Schur's characterization of multiplier sequences. As a limit case we obtain a new characterization of all log-concave sequences in terms of the zero location of certain associated polynomials. Our results also lead to an extension of Ruscheweyh's convolution lemma for functions which are analytic in the unit disk and to new necessary conditions for the validity of the Riemann Conjecture.2010 Mathematics Subject Classification. 30C10, 30C15, 26C10, 30D15, 05A99, 11M26.
We prove the case ρ = 1 4 of the following conjecture of Koumandos and Ruscheweyh: let s µ n (z) := n k=0 (µ) k k! z k , and for ρ ∈ (0, 1] let µ * (ρ) be the unique solution of, n ∈ N and z in the unit disk of C and µ * (ρ) is the largest number with this property. For the proof of this other new results are required that are of independent interest. For instance, we find the best possible lower bound µ 0 such that the derivative of x − Γ (x+µ) Γ (x+1) x 2−µ is completely monotonic on (0, ∞) for µ 0 ≤ µ < 1.
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