Applications of Certain Operators to the Classes of Analytic Functions Related to the Generalized Janowski Functions

We introduce certain subclasses of analytic functions related to the class of analytic, convex univalent functions. We discuss some results including inclusion relationships and invariance of the classes under convex convolution in terms of certain linear operators. Applications of these results associated with the generalized Janowski functions and conic domains are considered. Also, several radius problems are investigated.

On Fully-Convex Harmonic Functions and their Extension

‎Uniformly convex univalent functions that introduced by Goodman‎, ‎maps every circular arc contained in the open unit disk with center in it into a convex curve‎. ‎On the other hand‎, ‎a fully-convex harmonic function‎, ‎maps each subdisk $|z|=r<1$ onto a convex curve‎. ‎Here we synthesis these two ideas and introduce a family of univalent harmonic functions which are fully-convex and uniformly convex also‎. ‎In the following we will mention some examples of this subclass and obtain a necessary and sufficient conditions and finally a coefficient condition will attain with convolution‎.

Applications of differential subordinations for norm estimates of an integral operator

By considering the norm of a locally univalent function given bywe obtain such norm estimates for an operator of functions involving a convolution structure of convex univalent functions with a subclass of convex functions (defined by subordination). We also obtain some inequalities concerning this norm for functions under a certain fractional integral operator. Some implications of our results are briefly pointed out in the concluding section.

UNIVALENCE CONDITIONS FOR A NEW INTEGRAL OPERATOR

In the present paper, we will obtain norm estimates of the pre-Schwarzian derivatives for $F_{\lambda,\mu}(z)$, such that \[ F_{\lambda,\mu}(z) = \int_0^z \prod_{i=1}^{n} (f'_i(t))^{\lambda_i}\left( \frac{f_i(t)}{t} \right)^{\mu_i}dt \quad (z\in D),\] where $\lambda_i,\mu_i\in \mathbb{R}$, $\lambda_i=(\lambda_1,\lambda_2,\ldots,\lambda_n$, $\mu_i=(\mu_1,\mu_2,\ldots,\mu_n$ and $f_i$ belongs to the class of convex univalent functions $\mathcal{C}\subset \mathcal{S}$.

On Booth Lemniscate and Hadamard Product of Analytic Functions

In [RUSCHEWEYH, S.-SHEIL-SMALL, T.: Hadamard product of schlicht functions and the Poyla-Schoenberg conjecture, Comment. Math. Helv. 48 (1973), 119-135] the authors proved the P`olya-Schoenberg conjecture that the class of convex univalent functions is preserved under convolution, namely K ∗ K = K. They proved also that the class of starlike functions and the class of close-to-convex functions are closed under convolution with the class K. In this paper we consider similar convolution problems for some classes of functions. Especially we give a new applications of a result [SOKÓŁ, J.: Convolution and subordination in the convex hull of convex mappings, Appl. Math. Lett. 19 (2006), 303-306] on the subordinating relations in the convex hull of convex mappings under convolution. The paper deals with several ideas and techniques used in geometric function theory. Besides being an application to those results it provides interesting corollaries concerning special functions, regions and curves.