Let ${\mathcal A}$ be the class of functions analytic in the unit disk ${\mathbb D} := \{ z\in {\mathbb C}:\, |z| < 1 \}$ and normalized such that $f(z)=z+a_2z^2+a_3z^3+\cdots$. In this paper we study the class $\mathcal{U}(λ)$, $0<λ\leq1$, consisting of functions $f$ from ${\mathcal{A}}$ satisfying \[\left|\left(\frac{z}{f(z)}\right)^2f'(z)-1\right| < λ\quad (z\in {\mathbb D}).\] and give results regarding the Zalcman Conjecture, the generalised Zalcman conjecture, the Krushkal inequality and the second and third order Hankel determinant.
In this paper, subordination results are studied for certain subclass of p-valent meromorphic functions in the punctured unit disc having a pole of order p at the origin. The subclass under investigation is defined by using certain new linear operator. Moreover, we also introduced an interesting particular cases of these results in several corollaries.
In this article, we introduce a new family of sense preserving harmonic mappings f in the open unit disk and prove that functions in this family are close-to-convex. We give some basic properties such as coefficient bounds, growth estimates, convolution and determine the radius of convexity for the functions belonging to this family. In addition, we construct certain harmonic univalent polynomials belonging to this family.
The object of the present paper is to study certain properties and characteristics of the operator $Q_{p,β}^α$defined on p-valent analytic function by using technique of differential subordination.We also obtained result involving majorization problems by applying the operator to p-valent analytic function.Relevant connection of the the result are presented here with those obtained by earlier worker are pointed out.
In this paper we determine the disks $|z|<r\le1$ where for different classes of univalent functions, we have the property $${\rm Re}\left\{2\frac{zf'(z)}{f(z)}-\frac{z f''(z)}{f'(z)}\right\}>0\qquad (|z|<r).$$
In this paper, we determine necessary and sufficient conditions for the generalized Bessel function to be in certain subclasses of starlike and convex functions. Also, we obtain several corollaries as special cases of the main results, these corollaries gives the corresponding results of the familiar, modified and spherical Bessel functions.
In the present paper we introduce and investigate an interesting subclass K_{s}^{(k)}(γ,p) of analytic and p-valently close-to-convex functions in the open unit disk U. For functions belonging to this class, we derive several properties as the inclusion relationships and distortion theorems. The various results presented here would generalize many known recent results.
In this paper, we introduce and investigate two new subclasses of analytic functions in the open unit disk in the complex plane. Several interesting properties of the functions belonging to these classes are examined. Here, sufficient, and necessary and sufficient conditions for the functions belonging to these classes are also given. Moreover, coefficient bound estimates for the functions belonging to these classes are obtained.
In this article, we investigate some standard geometric properties of the integral operators $$ J_α[f](z)= \int_{0}^{z}\bigg(\frac{f(w)}{w}\bigg)^αdw, \,\,\, α\in \mathbb{C} \text{ and } |z|<1, $$ and $$ I_β[g](z)= \int_{0}^{z}\big(g'(w)\big)^βdw, \,\,\, β\in \mathbb{C} \text{ and } |z|<1, $$ where $f$ and $g$ are elements of certain classical families of normalized analytic functions defined on the unit disk. In particular, preserving properties of the Hornich sum of the operators $J_α$ and $I_β$ will be studied. Moreover, we also present sharp pre-Schwarzian norm estimate of such integrals.
In this paper, we determine the sharp estimates for Toeplitz determinants of a subclass of close-to-convex harmonic mappings. Moreover, we obtain an improved version of Bohr's inequalities for a subclass of close-to-convex harmonic mappings, whose analytic parts are Ma-Minda convex functions.
For analytic functions f(z) in the closed unit disk \bar{U}, two boundary points z_1 and z_2 such that α= (f'(z_1)+f'(z_2))/2 in f'(U) are considered. The object of the present paper is to discuss some interesting conditions for f(z) to be |f'(z)-1|<ρ|1-α| in U with some examples.
Let ${\mathcal U}^+$ be the class of analytic functions $f$ such that $\frac{z}{f(z)}$ has real and positive coefficients and $f^{-1}$ be its inverse. In this paper we give sharp estimates of the initial coefficients and initial logarithmic coefficients for $f$, as well as, sharp estimates of the second and the third Hankel determinant for $f$ and $f^{-1}$. We also show that the Zalcman conjecture holds for functions $f$ from ${\mathcal U}^+$.
In this paper, we continue our investigation of function spaces on certain classes of complex-valued functions. In particular, we give characterizations on Hardy-type, Bergman-type and Dirichlet-type spaces. Furthermore, we present applications of our results to certain nonlinear PDEs.
In the present investigation, we derive Fekete-Szegö inequality for the class $\mathcal{S}^α_{\mathscr{L}_{g}}(ϕ)$, introduced here. In addition to that, certain applications of our results are also discussed.
In this paper, four parameters Wright function is considered. Certain geometric properties such as starlikeness, convexity, uniform convexity and close-to-convexity are discussed for this function. Further, certain geometric properties of normalized Bessel function of the first kind and two parameters Wright function are studied as a consequence. Interesting corollaries and examples are provided to support that these results are better than the existing ones and improve several results available in the literature.
Some geometric properties of certain class of Le Roy type functions
Indian Institute of Technology Hyderabad（印度理工学院海德拉巴分校）；Shiv Nadar University（希夫纳达尔大学）；Dr. Shyama Prasad Mukherjee University（S.P.穆尔杰博士大学）；Indian Institute of Technology Madras（印度理工学院马德拉斯分校）
In this article, we wish to establish some first order differential subordination relations for certain Carathéodory functions with nice geometrical properties. Moreover, several implications are determined so that the normalized analytic function belongs to various subclasses of starlike functions.
We introduce the class of analytic functions $$\mathcal{F}(ψ):= \left\{f\in \mathcal{A}: \left(\frac{zf'(z)}{f(z)}-1\right) \prec ψ(z),\; ψ(0)=0 \right\},$$ where $ψ$ is univalent and establish the growth theorem with some geometric conditions on $ψ$ and obtain the Koebe domain with some related sharp inequalities. Note that functions in this class may not be univalent. As an application, we obtain the growth theorem for the complete range of $α$ and $β$ for the functions in the classes $\mathcal{BS}(α):= \{f\in \mathcal{A} : ({zf'(z)}/{f(z)})-1 \prec {z}/{(1-αz^2)},\; α\in [0,1) \}$ and $\mat…
In this paper we determine the region of variability for certain subclasses of univalent functions satisfying differential inequalities. In the final section we graphically illustrate the region of variability for several sets of parameters.
In the present work, we propose to investigate the Fekete-Szegö 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.
In this paper, we considered a generalized class of starlike functions defined by Kanas and Răducanu\cite{10} to obtain integral means inequalities and subordination results. Further, we obtain the for various subclasses of starlike functions.
Making use of Chebyshev polynomials, we obtain upper bound estimate for the second Hankel determinant of a subclass $\mathcal{N}_{σ}^μ\left( λ,t\right) $ of bi-univalent function class $σ.$
We present combinatorial and analytical results concerning a Sheffer sequence with a generating function of the form $G(x,z)=Q(z)^{x}Q(-z)^{1-x}$, where $Q$ is a quadratic polynomial with real zeros. By using the properties of Riordan matrices we address combinatorial properties and interpretations of our Sheffer sequence of polynomials and their coefficients. We also show that apart from two exceptional zeros, the zeros of polynomials with large enough degree in such a Sheffer sequence lie on the line $x=1/2+it$.
In this short note we employ the Briot-Bouquet differential subordination to determine the best possible inclusion relation within a certain family of analytic functions defined by the Salagean derivative.
We study the weak asymptotic behavior of the zeros of a family of a certain class of (generalized) hypergeometric polynomials, using the associated hypergeometric differential equation, as the parameters go to infinity. We describe the curve complex on which the zeros cluster, as level curves associated to integrals on an algebraic curve derived from the equation. In a certain degenerate case we make a precise conjecture, based and generalizing work by Duren, Driver et. al, and present experimental evidence for it.
We present combinatorial and analytical results concerning a Sheffer sequence with an exponential generating function of the form $G(s,z)=e^{czs+αz^{2}+βz^{4}}$, where $α, β, c \in \mathbb{R}$ with $β<0$ and $c\neq 0$. We demonstrate that the zeros of all polynomials in such a Sheffer sequence are either real, or purely imaginary. Additionally, using the properties of Riordan matrices we show that our Sheffer sequence satisfies a three-term recurrence relation of order 4, and we also exhibit a connection between the coefficients of these Sheffer polynomials and the number of nodes with a a giv…
Consider a scaled Nevanlinna-Pick interpolation problem and let $Π$ be the Blaschke product whose zeros are the nodes of the problem. It is proved that if $Π$ belongs to a certain class of inner functions, then the extremal solutions of the problem or most of them, are in the same class. Three different classical classes are considered: inner functions whose derivative is in a certain Hardy space, exponential Blaschke products and also the well known class of $α$-Blaschke products, for $0<α<1$.
Let $C$ be a smooth elliptic curve embedded in a smooth complex surface $X$ such that $C$ is a leaf of a suitable holomorphic foliation of $X$. We investigate complex analytic properties of a neighborhood of $C$ under some assumptions on complex dynamical properties of the holonomy function. As an application, we give an example of $(C, X)$ in which the line bundle $[C]$ is formally flat along $C$ however it does not admit a $C^\infty$ Hermitian metric with semi-positive curvature.
