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In 2003, Mercer presented an interesting variation of Jensen's inequality called Jensen–Mercer inequality for convex function. In the present paper, by employing harmonically convex function, we introduce analogous versions of Hermite–Hadamard inequalities of the Jensen–Mercer type via fractional integrals. As a result, we introduce several related fractional inequalities connected with the right and left differences of obtained new inequalities for differentiable harmonically convex mappings. As an application viewpoint, new estimates regarding hypergeometric functions and special means of real numbers are exemplified to determine the pertinence and validity of the suggested scheme. Our results presented here provide extensions of others given in the literature. The results proved in this paper may stimulate further research in this fascinating area. [ABSTRACT FROM AUTHOR]

In this work, we introduce the idea of n –polynomial harmonically s –type convex function. We elaborate the new introduced idea by examples and some interesting algebraic properties. As a result, new Hermite–Hadamard, some refinements of Hermite–Hadamard and Ostrowski type integral inequalities are established, which are the generalized variants of the previously known results for harmonically convex functions. Finally, we illustrate the applicability of this new investigation in special functions (hypergeometric function and special mean of real numbers). [ABSTRACT FROM AUTHOR]

For p-valently Janowski starlike and convex functions defined by applying subordination for the generalized Janowski function, the sharp upper bounds of a functional ∣ap+2 - μa2 p+1∣ related to the Fekete-Szeg¨o problem are given. [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]

The aim of the present paper is to study the p-valent analytic functions in the unit disk and satisfy the differential subordinations z(Ip)r, λ(f(z))(j+1)/(p - j)(ɔp(r, λ)f(z))(j) < (a + (aB + (A- B)β)z)/a(1 + Bz), where ɔp(r, λ) is an operator defined by Sălăgean and β is a complex number. Further we define a new related integral operator and also study the Fekete-Szego problem by proving some interesting properties. [ABSTRACT FROM AUTHOR]

Making use of the generalized hypergeometric functions, we define a new subclass of uniformly convex functions and a corresponding subclass of starlike functions with negative coefficients and obtain coefficient estimates, extreme points, the radii of close-to-convexity, starlikeness and convexity, and neighborhood results for the class TSml(α, β, γ). In particular, we obtain integral means inequalities for the function f that belongs to the class TSml(α, β, γ) in the unit disc. [ABSTRACT FROM AUTHOR]

The authors define a new subclass of A of functions involving complex order in the open unit disk U. For this new class, we obtain certain inclusion properties involving the Gaussian hypergeometric functions. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to give some characterizations for a (Gaussian) hypergeometric function to be in various subclasses of starlike and convex functions. We also consider an integral operator related to the hypergeometric function. [ABSTRACT FROM AUTHOR]