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We will study a maximal solution of the time-space fractional heat equation in complex domain. The fractional time is taken in the sense of the Riemann-Liouville operator, while the fractional space is assumed in the Srivastava-Owa operator. Here we employ some properties of the univalent functions in the unit disk to determine the upper bound of this solution. The maximal solution is illustrated in terms of the generalized hypergeometric functions. [ABSTRACT FROM AUTHOR]

This paper deals with recent developments of Hilbert-type discrete and integral inequalities by introducing kernels, weight functions, and multiparameters. Included are numerous generalizations, extensions, and refinements of Hilbert-type inequalities involving many special functions such as beta, gamma, logarithm, trigonometric, hyper-bolic, Bernoulli's functions and Bernoulli's numbers, Euler's constant, zeta function, and hypergeometric functions with many applications. Special attention is given to many equivalent inequalities and to conditions under which the constant factors involved in inequalities are the best possible. Many particular cases of Hilbert-type inequalities are presented with numerous applications. A large number of major books and recent research papers published during 2009-2012 are included to stimulate new interest in future study and research. [ABSTRACT FROM AUTHOR]

The aim of this research paper is to obtain explicit expressions of (1 - x)-2a 2F1[2b+ja,b ; -4x/(1x)2 ) for j = 0, ±1, ±2. For j = 0, we have the well-known transformation formula due to Gauss. The results are derived with the help of generalized Watson's theorem. Some known results obtained earlier follow special cases of our main findings. [ABSTRACT FROM AUTHOR]

We introduce and study a subclass ΣP (γ, k, λ, c) of meromorphic univalent functions defined by certain linear operator involving the generalized hypergeometric function. We obtain coefficient estimates, extreme points, growth and distortion inequalities, radii of meromorphic starlikeness, and convexity for the class ΣP (γ, k, λ, c) by fixing the second coefficient. Further, it is shown that the class ΣP (γ, k, λ) is closed under convex linear combination. [ABSTRACT FROM AUTHOR]

By making use of the Dziok-Srivastava operator, we introduce a new class of meromorphically multivalent functions. Some inclusion properties of functions belonging to this class are derived. [ABSTRACT FROM AUTHOR]

We introduce a new subclass of meromorphically analytic functions, which is defined by means of a Hadamard product (or convolution). A characterization property such as the coefficient bound is obtained for this class. The other related properties, which are investigated in this paper, include the distortion and the radius of starlikeness. We also consider several applications of our main results to the generalized hypergeometric functions. [ABSTRACT FROM AUTHOR]