For a given p-valent analytic function g with positive coefficients in the open unit disk ∆, we study a class of functions f(z) = z + ∑∞ n=m anz , an ≥ 0 satisfying 1 p < ( z(f ∗ g)′(z) (f ∗ g)(z) ) < α ( z ∈ ∆; 1 < α < m+ p 2p ) . Coefficient inequalities, distortion and covering theorems, as well as closure theorems are determined. The results obtained extend several known results as special ...

Lei Shi, Zhi-Gang Wang, and Jing-Ping Yi School of Mathematics and Statistics, Anyang Normal University, Henan, Anyang 455002, China Correspondence should be addressed to Lei Shi, shilei [email protected] Received 22 September 2012; Revised 26 October 2012; Accepted 31 October 2012 Academic Editor: Alberto Cabada Copyright q 2012 Lei Shi et al. This is an open access article distributed under the...

We give a combinatorial proof of the identity for the alternating convolution of the central binomial coefficients. Our proof entails applying an involution to certain colored permutations and showing that only permutations containing cycles of even length remain. The combinatorial identity

A sequence of coefficients appearing in a recurrence for the Narayana polynomials is generalized. The coefficients are given a probabilistic interpretation in terms of beta distributed random variables. The recurrence established by M. Lasalle is then obtained from a classical convolution identity. Some arithmetical properties of the generalized coefficients are also established.

Let Ap(p ∈ N) be the class of functions f(z) = z + ∑∞ m=1 ap+mz p+m which are analytic in the unit disk. By virtue of the Ruscheweyh derivatives we introduce the new subclasses Cp(n, α, β, λ, μ) of Ap. Subordination relations, inclusion relations, convolution properties and a sharp coefficient estimate are obtained. We also give a sufficient condition for a function to be in Cp(n, α, β, λ, μ).

In this study we extend the classification of partial differential equations to the further using the convolutions products. The purpose of this study is to compute the solutions of some explicit initial-boundary value problems for one-dimensional wave equation with variable coefficients by means of Laplace transform which in general has no solution.

Sharp bounds for japþ2 lapþ1j and jap+3j are derived for certain p-valent analytic functions. These are applied to obtain Fekete-Szegö like inequalities for several classes of functions defined by convolution. 2006 Elsevier Inc. All rights reserved.

For certain meromorphic function g and h, we study a class of functions f(z) = z−1 + ∑∞ n=1 fnz , (fn ≥ 0), defined in the punctured unit disk ∆∗, satisfying < ( (f ∗ g)(z) (f ∗ h)(z) ) > α (z ∈ ∆; 0 ≤ α < 1) . Coefficient inequalities, growth and distortion inequalities, as well as closure results are obtained. Properties of an integral operator and its inverse defined on the new class is also...

In this paper we will prove bounds for the fourth power moment in the t aspect over a short interval of automorphic L-functions L(s, g) for GL(2) on the central critical line Re s = 1/2. Here g is a fixed holomorphic or Maass Hecke eigenform for the modular group SL2(Z), or in certain cases, for the Hecke congruence subgroup Γ0(N ) with N > 1. The short interval is from a large K to K + K103/13...

The purpose of the present paper is to establish some results involving coefficient conditions, distortion bounds, extreme points, convolution, convex combinations and neighborhoods for a new class of harmonic univalent functions in the open unit disc. We also discuss a class preserving integral operator. Relevant connections of the results presented here with various known results are briefly ...