Harmonic Convex Functions Research Articles

Hermite-Hadamard and Simpson-Like Type Inequalities for Differentiable Harmonically Convex Functions

A new identity for differentiable functions is derived. A consequence of the identity is that the author establishes some new general inequalities containing all of the Hermite-Hadamard and Simpson-like types for functions whose derivatives in absolute value at certain power are harmonically convex. Some applications to special means of real numbers are also given.

Radius of convexity of partial sums of functions in the close-to-convex family

Let F denote the class of all normalized analytic functions f that are locally univalent in the unit disk |z|<1 satisfying the condition Re(1+zf″(z)f′(z))>−12 for |z|<1. Functions in F are known to be close-to-convex (univalent) in the unit disk. This class plays a crucial role in the discussion on certain extremal problems for the class of complex-valued and sense-preserving harmonic convex functions and some other related problems in determining univalence criteria for sense-preserving harmonic mappings. In this article, we show that every section of a function in the class F is convex in the disk |z|<1/6. The radius 1/6 is best possible. We conjecture that every section of functions in the family F is univalent and close-to-convex in the disk |z|<1/3.

Radius of close-to-convexity and fully starlikeness of harmonic mappings

Let denote the class of all normalized complex-valued harmonic functions in the unit disk , and let denote the class of univalent and sense-preserving functions in such that . If denotes the harmonic Koebe function whose dilation is , then and it is conjectured that is extremal for the coefficient problem in . If the conjecture were true, then contains the family , where Here, and denote the Maclaurin coefficients of and . We show that the radius of univalence of the family is . We also show that this number is also the radius of the fully starlikeness of . Analogous results are proved for a family which contains the class of harmonic convex functions in . We use the new coefficient estimate for bounded harmonic mappings and Lemma 1.6 to improve Bloch-Landau constant for bounded harmonic mappings.

On a Subclass of Harmonic Convex Functions of Complex Order

We introduce and study a subclass of harmonic convex functions of complex order. Coefficient bounds, extreme points, distortion bounds, convolution conditions, and convex combination are determined for functions in this class. Further, we obtain the closure property of this class under integral operator.

ON A SUBCLASS OF CERTAIN CONVEX HARMONIC FUNCTIONS

We deflne and investigate a subclass of complex val- ued harmonic convex functions that are univalent and sense pre- serving in the open unit disk. We obtain coe-cient conditions, extreme points, distortion bounds, convolution conditions for the above family of harmonic functions.

Generalization of convex and related functions

In this paper some new kinds of generalized Logarithmic and Harmonic Convex functions have been introduced and their relationships with known concepts have been discussed.