Coefficient bounds for a subclass of starlike functions of complex order

In the present work, we determine coefficient bounds for functions in certain subclass of starlike functions of complex order b, which are introduced here by means of a multiplier differential operator. Several corollaries and consequences of the main results are also considered.     

Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions

Estimates on the initial coefficients are obtained for normalized analytic functions f in the open unit disk with f and its inverse g=f⁻¹ satisfying the conditions that zf^′(z)/f(z) and zg^′(z)/g(z) are both subordinate to a univalent function whose range is symmetric with respect to the real axis. Several related classes of functions are also considered, and connections to earlier known results are made.     

Coefficient bounds for some families of starlike and convex functions of complex order

In the present work, the authors determine coefficient bounds for functions in certain subclasses of starlike and convex functions of complex order, which are introduced here by means of a family of nonhomogeneous Cauchy–Euler differential equations. Several corollaries and consequences of the main results are also considered.     

Duality techniques for certain integral transforms to be starlike

For β<1, let denote the class of all normalized analytic functions f such that

     

Norm estimates for convolution transforms of certain classes of analytic functions

For normalized analytic functions f in the unit disk Δ, we consider the class

     

Norm estimates for the Alexander transforms of convex functions of order alpha

The hyperbolic sup norm of the pre-Schwarzian derivative of a locally univalent function on the unit disk measures the deviation of the function from similarities. We present sharp norm estimates for the Alexander transforms of convex functions of order α, 0?α<1.     

Coefficient estimates of negative powers and inverse coefficients for certain starlike functions

For ?1≤B<A≤1, let \(\mathcal {S}^{*}(A,B)\) denote the class of normalized analytic functions \(f(z)= z+{\sum }_{n=2}^{\infty }a_{n} z^{n}\) in |z|<1 which satisfy the subordination relation z f ^′(z)/f(z)?(1 + A z)/(1 + B z) and Σ^?(A,B) be the corresponding class of meromorphic functions in |z|>1. For \(f\in \mathcal {S}^{*}(A,B)\) and λ>0, we shall estimate the absolute value of the Taylor coefficients a _n (?λ,f) of the analytic function (f(z)/z)^?λ . Using this we shall determine the coefficient estimate for inverses of functions in the classes \(\mathcal {S}^{*}(A,B)\) and Σ^?(A,B).     

Some criteria for meromorphic multivalent starlike functions

The main purpose of the present paper is to derive some new criteria for meromorphic multivalent starlike functions.     

Landau’s theorem for functions with logharmonic Laplacian

In this paper, we show the existence of Landau constant for functions with logharmonic Laplacian of the form F(z) = ∣z∣²L(z) + K(z), ∣z∣ < 1, where L is logharmonic and K is harmonic. Moreover, the problem of minimizing the area is solved     

Starlike approximations of analytic functions

When an analytic function is not univalent, it is often of interest to approximate it by univalent functions. In this paper we introduce a measure of the non-univalency of a function and we derive a method for constructing the best starlike univalent approximations of analytic functions with respect to it, suitable for both practical problems and numerical implementation.     

Coefficient inequalities for a subclass of starlike functions

A function f(z) = z ? ∑^∞_{n = 2}a_nzⁿ, a_n ? 0, analytic and univalent in the unit disk, is said to be in the family $\text{T}{}^1\text{(a, b)}$, a real and b ? 0, if $\text{¦(}\text{zf′}\text{f}\text{) ? a¦ ? b}$ for all z in the unit disk. A complete characterization is found for $\text{T}{}^1\text{(a, b)}$ when a ? 1. Also, sharp coefficient bounds are determined for certain subclasses of $\text{T}{}^1\text{(a, b)}$ when a < 1; however, examples are given to show that these bounds do not remain valid for the whole family.     

Region of variability of two subclasses of univalent functions

Let F₁ (F₂ respectively) denote the class of analytic functions f in the unit disk |z|<1 with f(0)=0=f^′(0)−1 satisfying the condition RePf(z)<3/2 (RePf(z)>−1/2 respectively) in |z|<1, where Pf(z)=1+zf^″(z)/f^′(z). For any fixed z₀ in the unit disk and λ∈[0,1), we shall determine the region of variability for logf^′(z₀) when f ranges over the class and , respectively.     

Some starlikeness criterions for analytic functions

We determine a condition on M, α, λ and μ for which

     

On the univalence of functions with logharmonic laplacian

In this paper, we prove Radó’s theorem holds for functions of the form is logharmonic. We show that if F is of the form , where is logharmonic, then F is starlike iff ψ(z)=h(z)/g(z) is starlike. In addition, when , where L is logharmonic and H is harmonic, we give the sufficient conditions for F to be locally univalent.     

Upper and lower estimates for positive solutions of the higher order Lidstone boundary value problem

We consider the higher order Lidstone boundary value problem. New upper and lower estimates for positive solutions of the problem are obtained. A discussion of the sharpness of the estimates is included.     

An Inverse Coefficient Problem for an Integro-differential Equation

In this article we consider the inverse coefficient problem of recovering the function { ( x ) system of partial differential equations that can be reduced to a second order integro-differential equation $ -u_{xx} + c(x)u_{x} + d\phi (x)u-\gamma d\phi (x)\int _{0}^{t}e^{-\gamma (t-\tau )}u(x,\tau )\, d\tau = 0 $ with boundary conditions. We prove the existence and uniqueness of solutions to the inverse problem and develop a numerical algorithm for solving this problem. Computational results for some examples are presented.     

Asymptotic properties of some functions of nonparametric estimates of a density function

Let be a nonparametric estimate of a two-dimensional density f(x,y) constructed with the help of two-dimensional “window.” The main purpose of the paper is to study the asymptotic properties of the marginal moments estimates and differentiable functions , of these moments.