The objective of this paper is to obtain sharp upper bound for the function f for the second Hankel determinant |a2a4 − a 2 3 |, when it belongs to the class of functions whose derivative has a positive real part of order α (0 ≤ α < 1), denoted by RT (α). Further, an upper bound for the inverse function of f for the nonlinear functional (also called the second Hankel functional), denoted by |t2t4 − t 2 3 |, was determined when it belongs to the same class of functions, using Toeplitz determinants.
The spaces of entire functions represented by Dirichlet series have been studied by Hussein and Kamthan and others. In the present paper we consider the space X of all entire functions defined by vector-valued Dirichlet series and study the properties of a sequence space which is defined using the type of an entire function represented by vectorvalued Dirichlet series. The main result concerns with obtaining the nature of the dual space of this sequence space and coefficient multipliers for some classes of vector-valued Dirichlet series.
In the present paper, we investigate certain geometric properties and inequalities for the Wright function and mention a few important consequences of our main results. A nonlinear differential equation involving the Wright function is also investigated.
We generalize some criteria of boundedness of L-index in joint variables for in a bidisc analytic functions. Our propositions give an estimate the maximum modulus on a skeleton in a bidisc and an estimate of (p + 1)th partial derivative by lower order partial derivatives (analogue of Hayman’s theorem).
H. Silverman (1999) investigated the properties of functions defined in terms of the quotient of the analytic representations of convex and starlike functions. Many research workers have been working on analytic functions to be strongly starlike like Obradovi´c and Owa (1989), Takahashi and Nunokawa (2003), Lin (1993) etc. In this paper we obtain a sufficient condition for p-valent functions to be strongly starlike of order α.
We develop a theory of removable singularities for the weighted Bergman space ${\mathcal A}^p_\mu (\Omega )=\lbrace f \text{analytic} \text{in} \Omega \: \int _\Omega |f|^p \mathrm{d}\mu < \infty \rbrace $, where $\mu $ is a Radon measure on $\mathbb{C}$. The set $A$ is weakly removable for ${\mathcal A}^p_\mu (\Omega \setminus A)$ if ${\mathcal A}^p_\mu (\Omega \setminus A) \subset \text{Hol}(\Omega )$, and strongly removable for ${\mathcal A}^p_\mu (\Omega \setminus A)$ if ${\mathcal A}^p_\mu (\Omega \setminus A) = {\mathcal A}^p_\mu (\Omega )$. The general theory developed is in many ways similar to the theory of removable singularities for Hardy $H^p$ spaces, $\mathop {\mathrm BMO}$ and locally Lipschitz spaces of analytic functions, including the existence of counterexamples to many plausible properties, e.g. the union of two compact removable singularities needs not be removable. In the case when weak and strong removability are the same for all sets, in particular if $\mu $ is absolutely continuous with respect to the Lebesgue measure $m$, we are able to say more than in the general case. In this case we obtain a Dolzhenko type result saying that a countable union of compact removable singularities is removable. When $\mathrm{d}\mu = w\mathrm{d}m$ and $w$ is a Muckenhoupt $A_p$ weight, $1<p<\infty $, the removable singularities are characterized as the null sets of the weighted Sobolev space capacity with respect to the dual exponent $p^{\prime }=p/(p-1)$ and the dual weight $w^{\prime }=w^{1/(1-p)}$.
Due to the fact that in the case $q>1$ the $q$-Bernstein polynomials are no longer positive linear operators on $C[0,1],$ the study of their convergence properties turns out to be essentially more difficult than that for $q<1.$ In this paper, new saturation theorems related to the convergence of $q$-Bernstein polynomials in the case $q>1$ are proved.