Estimates for the hyperbolic and quasihyperbolic metrics in hyperbolic regions

In this paper we consider ordinary derivative of universal covering mappings $f$ of hyperbolic regions $D$ in the complex plane. We obtain sharp bounds for the ratio $|f'(z)|/{\rm dist}(f(z),\partial f(D))$ in terms of the hyperbolic density in simply connection domains. In arbitrary domains, we find a necessary and sufficient condition for an upper bound for the quantity $|f'(z)|/{\rm dist}(f(z),\partial f(D))$ to hold in terms of the hyperbolic density. As an application of the above results, it is observed that the bounds for the quantity of the above type are closely connected with similar bounds for $|f''(z)/f'(z)|$.Comment: 8 page

Properties of β\beta-Ces\`aro operators on α\alpha-Bloch space

For each $\alpha > 0$, the $\alpha$-Bloch space is consisting of all analytic functions $f$ on the unit disk satisfying $\sup_{|z|<1} (1-|z|^2)^\alpha |f'(z)| < + \infty.$ In this paper, we consider the following complex integral operator, namely the $\beta$-Ces\`{a}ro operator \begin{equation} C_\beta(f)(z)=\int_{0}^{z}\frac{f(w)}{w(1-w)^{\beta}}dw \nonumber \end{equation} and its generalization, acting from the $\alpha$-Bloch space to itself, where $f(0)=0$ and $\beta\in\mathbb{R}$. We investigate the boundedness and compactness of the $\beta$-Ces\`{a}ro operators and their generalization. Also we calculate the essential norm and spectrum of these operators.Comment: 24 pages, Rocky Mountain Journal of Mathematics (to appear

Meromorphic functions with small Schwarzian derivative

We consider the family of all meromorphic functions $f$ of the form $$f(z)=\frac{1}{z}+b_0+b_1z+b_2z^2+\cdots$$ analytic and locally univalent in the puncture disk $\mathbb{D}_0:=\{z\in\mathbb{C}:\,0<|z|<1\}$. Our first objective in this paper is to find a sufficient condition for $f$ to be meromorphically convex of order $\alpha$, $0\le \alpha<1$, in terms of the fact that the absolute value of the well-known Schwarzian derivative $S_f (z)$ of $f$ is bounded above by a smallest positive root of a non-linear equation. Secondly, we consider a family of functions $g$ of the form $g(z)=z+a_2z^2+a_3z^3+\cdots$ analytic and locally univalent in the open unit disk $\mathbb{D}:=\{z\in\mathbb{C}:\,|z|<1\}$, and show that $g$ is belonging to a family of functions convex in one direction if $|S_g(z)|$ is bounded above by a small positive constant depending on the second coefficient $a_2$. In particular, we show that such functions $g$ are also contained in the starlike and close-to-convex family.Comment: 16 pages. Submitted to a journa

Radius of convexity of partial sums of odd functions in the close-to-convex family

We consider the class of all analytic and locally univalent functions $f$ of the form $f(z)=z+\sum_{n=2}^\infty a_{2n-1} z^{2n-1}$, $|z|<1$, satisfying the condition $${\rm Re}\,\left(1+\frac{zf^{\prime\prime}(z)}{f^\prime (z)}\right)>-\frac{1}{2}.$$ We show that every section $s_{2n-1}(z)=z+\sum_{k=2}^na_{2k-1}z^{2k-1}$, of $f$, is convex in the disk $|z|<\sqrt{2}/3$. We also prove that the radius $\sqrt{2}/3$ is best possible, i.e. the number $\sqrt{2}/3$ cannot be replaced by a larger one.Comment: 12 pages, 3 figure

On coefficient functionals associated with the Zalcman conjecture

We consider certain subfamilies, of the family of univalent functions in the open unit disk, defined by means of sufficient coefficient conditions for univalency. This article is devoted to studying the problem of the well-known conjecture of Zalcman consisting of a generalized coefficient functional, the so-called generalized Zalcman conjecture problem, for functions belonging to those subfamilies. We estimate the bounds associated with the generalized coefficient functional and show that the estimates are sharp.Comment: 8 page

Interpolation on Gauss hypergeometric functions with an application

In this paper, we use some standard numerical techniques to approximate the hypergeometric function $${}_2F_1[a,b;c;x]=1+\frac{ab}{c}x+\frac{a(a+1)b(b+1)}{c(c+1)}\frac{x^2}{2!}+\cdots$$ for a range of parameter triples $(a,b,c)$ on the interval $0<x<1$. Some of the familiar hypergeometric functional identities and asymptotic behavior of the hypergeometric function at $x=1$ play crucial roles in deriving the formula for such approximations. We also focus on error analysis of the numerical approximations leading to monotone properties of quotient of gamma functions in parameter triples $(a,b,c)$. Finally, an application to continued fractions of Gauss is discussed followed by concluding remarks consisting of recent works on related problems.Comment: To appear in Involve-A Journal of Mathematics, 16 page

A Gromov hyperbolic metric vs the hyperbolic and other related metrics

We mainly consider two metrics: a Gromov hyperbolic metric and a scale invariant Cassinian metric. We compare these two metrics and obtain their relationship with certain well-known hyperbolic-type metrics, leading to several inclusion relations between the associated metric balls.Comment: 20 pages, submitted to a journa

Successive coefficients for spirallike and related functions

We consider the family of all analytic and univalent functions in the unit disk of the form $f(z)=z+a_2z^2+a_3z^3+\cdots$. Our objective in this paper is to estimate the difference of the moduli of successive coefficients, that is $\big | |a_{n+1}|-|a_n|\big |$, for $f$ belonging to the family of $\gamma$-spirallike functions of order $\alpha$. Our particular results include the case of starlike and convex functions of order $\alpha$ and other related class of functions.Comment: 12 pages, Rev. R. Acad. Cienc. Exactas F\'is. Nat. Ser. A Mat. RACSAM. To appea

Carath\'eodory density of the Hurwitz metric on plane domains

It is well-known that the Carath\'eodory metric is a natural generalization of the Poincar\'e metric, namely, the hyperbolic metric of the unit disk. In 2016, the Hurwitz metric was introduced by D. Minda in arbitrary proper subdomains of the complex plane and he proved that this metric coincides with the hyperbolic metric when the domains are simply connected. In this paper, we define a new metric which generalizes the Hurwitz metric in the sense of Carath\'eodory. Our main focus is to study its various basic properties in connection with the Hurwitz metric.Comment: 10 pages, Bull. Malay Math. Sci. Soc., To appea

Bohr inequalities for certain integral operators

In this article, we determine sharp Bohr-type radii for certain complex integral operators defined on a set of bounded analytic functions in the unit disk.Comment: 10 pages; Submitted to a journa