Related papers: Integral transform for Logharmonic mappings
We show that complex geometric features of Teichmuller spaces create explicitly the extremals of generic homogeneous holomorphic functionals on univalent functions. In particular this gives proofs of the well-known Zalcman and Bieberbach…
The aim of this paper is to obtain the Schwarz-Pick type inequality for $\alpha$-harmonic functions $f$ in the unit disk and get estimates on the coefficients of $f$. As an application, a Landau type theorem of $\alpha$-harmonic functions…
A new class of integrals involving the confluent hypergeometric function ${}_1F_{1}(a;c;z)$ and the Riemann $\Xi$-function is considered. It generalizes a class containing some integrals of S. Ramanujan, G.H. Hardy and W.L. Ferrar and gives…
We prove the integrality of the Taylor coefficients of roots of mirror maps at the origin. By mirror maps, we mean formal power series z.exp(G(z)/F(z)), where F(z) and G(z)+log(z)F(z) are particular solutions of certain generalized…
The need to evaluate Logarithmic integrals is ubiquitous in essentially all quantitative areas including mathematical sciences, physical sciences. Some recent developments in Physics namely Feynman diagrams deals with the evaluation of…
We study the family of Fourier-Laplace transforms $$ F_{\alpha,\beta}(z)= \operatorname*{F.p.} \int_{0}^{\infty} t^{\beta}\exp(\mathrm{i} t^{\alpha}-\mathrm{i} z t)\:\mathrm{d} t, \quad \operatorname*{Im} z<0, $$ for $\alpha>1$ and…
Let ${\mathcal S}$ denote the set of all univalent analytic functions $f(z)=z+\sum_{n=2}^{\infty}a_n z^n$ on the unit disk $|z|<1$. In 1946 B. Friedman found that the set $\mathcal S$ of those functions which have integer coefficients…
We consider the convolution of half-plane harmonic mappings with respective dilatations $(z+a)/(1+az)$ and $e^{i\theta}z^{n}$, where $-1<a<1$ and $\theta\in\mathbb{R},n\in\mathbb{N}$. We prove that such convolutions are locally univalent…
The Schwarz lemma as one of the most influential results in complex analysis and it has a great impact to the development of several research fields, such as geometric function theory, hyperbolic geometry, complex dynamical systems, and…
Since the nonlinear integral transforms $J_{\alpha}[f](z) = \int_{0}^{z}(f'(u))^{\alpha} du$ and $I_{\alpha}[f](z) =\int_0^z (f(u)/u)^{\alpha} du$ with a complex number $\alpha$ have been introduced, a great number of studies were dedicated…
In this study, we establish certain Landau-type theorems 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, with $L$ and $K$ having bounded length distortion in…
Emil Artin defined a zeta function for algebraic curves over finite fields and made a conjecture about them analogous to the famous Riemann hypothesis. This and other conjectures about these zeta functions would come to be called the Weil…
The purpose of this paper is to make an introduction to univalent function theory for readers of any level, assuming only foundational knowledge in real and complex analysis. In particular, we state and proof (with details) important…
The aim of this paper is to establish properties of the solutions to the $\alpha$-harmonic equations: $\Delta_{\alpha}(f(z))=\partial{z}[(1-{|{z}|}^{2})^{-\alpha} \overline{\partial}{z}f](z)=g(z)$, where…
Using Jack polynomials, Goulden and Jackson have introduced a one parameter deformation $\tau_b$ of the generating series of bipartite maps, which generalizes the partition function of $\beta$-ensembles of random matrices. The Matching-Jack…
The Cauchy integral formula in Clifford analysis allows us to associate a holomorphic function $\tilde f:L_n\to \C$ on the Lie ball $L_n$ in $\C^n$ with its monogenic counterpart $f:B_1(0)\to \C^{n+1}$ via the formula $\tilde f(z) =…
The algebra B of bicomplex numbers is viewed as a complexification of the Archimedean f-algebra of hyperbolic numbers D. This lattice-theoretic approach allows us to establish new properties of the so-called D-norms. In particular, we show…
We use the Baernstein star-function to investigate several questions about the integral means of the convolution of two analytic functions in the unit disc. The theory of univalent functions plays a basic role in our work.
The classical estimate of Bieberbach -- that $|a_2|\le2$ for a given univalent function $\phi(z)=z+a_2z^2+...$ in the class $S$ -- leads to best possible pointwise estimates of the ratio $\phi''(z)/\phi'(z)$ for $\phi\in S$, first obtained…
We give a necessary and sufficient condition for the integrality of the Taylor coefficients of mirror maps at the origin. By mirror maps, we mean formal power series z.exp(G(z)/F(z)), where F(z) and G(z)+log(z)F(z) are particular solutions…