Coefficient and Fekete-Szeg\"o problem estimates for certain subclass of analytic and bi-univalent functions

In this paper, we obtain the Fekete-Szeg\"{o} problem for the $k$-th $(k\geq1)$ root transform of the analytic and normalized functions $f$ satisfying the condition \begin{equation*} 1+\frac{\alpha-\pi}{2 \sin \alpha}< {\rm Re}\left\{\frac{zf'(z)}{f(z)}\right\} < 1+\frac{\alpha}{2\sin \alpha} \quad (|z|<1), \end{equation*} where $\pi/2\leq \alpha<\pi$. Afterwards, by the above two-sided inequality we introduce and investigate a certain subclass of analytic and bi-univalent functions in the disk $|z|<1$ and obtain upper bounds for the first few coefficients and Fekete-Szeg\"{o} problem for functions belonging to this analytic and bi-univalent function class.Comment: 9 page

New Subclass of Pseudo-type Meromorphic Bi-Univalent functions of complex order

In the present article, we define a new subclass of pseudo-type meromorphic bi-univalent functions class $\Sigma'$ of complex order $\gamma \in \mathbb{C}\backslash \{0\}$ and investigate the initial coefficient estimates $|b_0|, |b_1|$ and $|b_2|.$ Further we mention several new or known consequences of our result.Comment: arXiv admin note: text overlap with arXiv:1108.4087, arXiv:1108.4089 by other author

The Fekete-Szego Coefficient Inequality For a New Class of m-Fold Symmetric Bi-Univalent Functions Satisfying Subordination Condition

In this paper, we investigate a new subclass of analytic and m-fold symmetric bi-univalent functions satisfying subordination in the open unit disk U. We consider the Fekete-Szeg\"o inequalities for this class. Also, we establish estimates for the coefficients for this subclas and several related classes are also considered and connections to earlier known results are made.Comment: 16 pages. arXiv admin note: substantial text overlap with arXiv:1603.01120 by other author

Coefficients of the Inverse Functions and Radius Estimates of Certain Starlike Functions

Ma-Minda class (of starlike functions) consists of all normalized analytic functions $f$ on the unit disk for which the image of $zf'(z)/f(z)$ is contained in the some starlike region in the right-half plane. We obtain the best possible bounds on the second and third coefficient for the inverse functions of functions in the Ma-Minda class. The bounds on the Fekete-Szeg\"o functional and the second Hankel determinant of the inverse functions of the functions belonging to the Ma-Minda class are also determined. Further, the bounds on the first five coefficients of the inverse functions are investigated for two particular subclasses of the Ma-Minda class. In addition, some radius estimates associated with the two subclasses are also computed

On third Hankel determinants for subclasses of analytic functions and close-to-convex harmonic mappings

In this paper, we obtain the upper bounds to the third Hankel determinants for starlike functions of order $\alpha$, convex functions of order $\alpha$ and bounded turning functions of order $\alpha$. Furthermore, several relevant results on a new subclass of close-to-convex harmonic mappings are obtained. Connections of the results presented here to those that can be found in the literature are also discussed.Comment: 13 page

A subclass of harmonic univalent mappings with a restricted analytic part

In this article, a subclass of univalent harmonic mapping is introduced by restricting its analytic part to lie in the class $\mathcal{S}^{\delta}[\alpha]$, $0\leq \alpha < 1$, $-\infty < \delta < \infty$ which has been introduced and studied by Kumar \cite{Kumar87} (see also \cite{Mishra95}, \cite{MishraChoudhury95}, \cite{MishraDas96}, \cite{MishraGochhayat06}). Coefficient estimations, growth and distortion properties, area theorem and covering estimates of functions in the newly defined class have been established. Furthermore, we also found bounds for the Bloch's constant for all functions in that family.Comment: 12 pages

Fekete-Szeg\"o problem for certain classes of Ma-Minda bi-univalent functions

In the present work, we propose to investigate the Fekete-Szeg\"o inequalities certain classes of analytic and bi-univalent functions defined by subordination. The results in the bounds of the third coefficient which improve many known results concerning different classes of bi-univalent functions. Some interesting applications of the results presented here are also discussed.Comment: 8 Page

Absolutely Convex, Uniformly Starlike and Uniformly Convex Harmonic Mappings

In this paper, we prove necessary and sufficient conditions for a sense-preserving harmonic function to be absolutely convex in the open unit disk. We also estimate the coefficient bound and obtain growth, covering and area theorems for absolutely convex harmonic mappings. A natural generalization of the classical Bernardi type operator for harmonic functions is considered and its connection between certain classes of uniformly starlike harmonic functions and uniformly convex harmonic functions is also investigated. At the end, as applications, we present a number of results connected with hypergeometric and polylogarithm functions.Comment: 16 pages; A version of it is appear in the journal: Complex Variables and Elliptic Equation

Estimates for the Initial Coefficients of Bi-univalent Functions

A bi-univalent function is a univalent function defined on the unit disk with its inverse also univalent on the unit disk. Estimates for the initial coefficients are obtained for bi-univalent functions belonging to certain classes defined by subordination and relevant connections with earlier results are pointed out.Comment: 12 page

Some inequalities for a certain subclass of starlike functions

In 2011, Sok\'{o}{\l} (Comput. Math. Appl. 62, 611--619) introduced and studied the class $\mathcal{SK}(\alpha)$ as a certain subclass of starlike functions, consists of all functions $f$ ($f(0)=0=f'(0)-1$) which satisfy in the following subordination relation: \begin{equation*} \frac{zf'(z)}{f(z)}\prec \frac{3}{3+(\alpha-3)z-\alpha z^2} \qquad |z|<1, \end{equation*} where $-3<\alpha\leq1$. Also, he obtained some interesting results for the class $\mathcal{SK}(\alpha)$. In this paper, some another properties of this class, including infimum of $\mathfrak{Re}\frac{f(z)}{z}$, order of strongly starlikeness, the sharp logarithmic coefficients inequality and the sharp Fekete-Szeg\"{o} inequality are investigated.Comment: 10 pages, 1 figur