A continuous function f = u + iv is a complex valued harmonic function in a complex domain C if both u and v are real harmonic in C. In any simply connected domain D ⊂ C we can write f(z) = h + g, where h and g are analytic in D. We call h the analytic part and g the co-analytic part of f . A necessary and sufficient condition for f to be locally univalent and sense-preserving in D is that |h′(...

Recently, many papers in the theory of univalent functions have been devoted to mapping and characterization properties of various linear integral or integro-differential operators in the class S (of normalized analytic and univalent functions in the open unit disk U), and in its subclasses (as the classes S∗ of the starlike functions and K of the convex functions in U). Among these operators, ...

In this paper, we have introduced and investigated two new subclasses of the function class Δ of bi-univalent functions defined in the open unit disk, which are associated with the generalized Hypergeometric function. Furthermore, we find estimates on the Taylor-Maclaurin coefficient | a2 | and | a3 | for the functions belonging to these new classes.

A continuous function f u iv is a complex-valued harmonic function in a complex domain Ω if both u and v are real and harmonic inΩ. In any simply connected domainD ⊂ Ω, we can write f h g, where h and g are analytic inD. We call h the analytic part and g the coanalytic part of f . A necessary and sufficient condition for f to be locally univalent and orientation preserving in D is that |h′ z | ...

Let φ (z) be a fixed analytic and univalent function of the form φ(z) = z + ∑∞ k=2 ckz k and Hφ (ck, δ) be the subclass consisting of analytic and univalent functions f of the form f(z) = z + ∑∞ k=2 akz k which satisfy the inequality ∑∞ k=2 ck |ak| ≤ δ. In this paper, we determine the sharp lower bounds for Re { Dpf(z) Dfn(z) } and Re { Dfn(z) Dpf(z) } , where fn(z) = z + ∑n k=2 akz k be the se...

Ruscheweyh and Sheil-Small proved that convexity is preserved under the convolution of univalent analytic mappings in K. However, when we consider the convolution of univalent harmonic convex mappings in K H , this property does not hold. In fact, such convolutions may not be univalent. We establish some results concerning the convolution of univalent harmonic convex mappings provided that it i...

Recently, q-analogue of Noor integral operator and other special operators became importance in the field Geometric Function Theory. In this study, by connecting principle subordination we introduced an interesting class bi-univalent functions obtained coefficient estimates for new class.