• Title, Summary, Keyword: radius of starlikeness

Search Result 14, Processing Time 0.039 seconds

PRODUCT AND CONVOLUTION OF CERTAIN UNIVALENT FUNCTIONS

  • Jain, Naveen Kumar;Ravichandran, V.
    • Honam Mathematical Journal
    • /
    • v.38 no.4
    • /
    • pp.701-724
    • /
    • 2016
  • For $f_i$ belonging to various subclasses of univalent functions, we investigate the product given by $h(z)=z{\prod_{i=1}^{n}}(f_i(z)/z)^{{\gamma}_i}$.The largest radius ${\rho}$ is determined such that $h({\rho}z)/{\rho}$ is starlike of order ${\beta}$, $0{\leq}{\beta}$ < 1 or to belong to other subclasses of univalent functions. We also determine the sharp radius of starlikeness of order ${\beta}$and other radius for the convolution f*g of two starlike functions f, g.

RADII PROBLEMS FOR THE GENERALIZED MITTAG-LEFFLER FUNCTIONS

  • Prajapati, Anuja
    • Journal of the Korean Mathematical Society
    • /
    • v.57 no.4
    • /
    • pp.1031-1052
    • /
    • 2020
  • In this paper our aim is to find various radii problems of the generalized Mittag-Leffler function for three different kinds of normalization by using their Hadamard factorization in such a way that the resulting functions are analytic. The basic tool of this study is the Mittag-Leffler function in series. Also we have shown that the obtained radii are the smallest positive roots of some functional equations.

RADIUS OF FULLY STARLIKENESS AND FULLY CONVEXITY OF HARMONIC LINEAR DIFFERENTIAL OPERATOR

  • Liu, ZhiHong;Ponnusamy, Saminathan
    • Bulletin of the Korean Mathematical Society
    • /
    • v.55 no.3
    • /
    • pp.819-835
    • /
    • 2018
  • Let $f=h+{\bar{g}}$ be a normalized harmonic mapping in the unit disk $\mathbb{D}$. In this paper, we obtain the sharp radius of univalence, fully starlikeness and fully convexity of the harmonic linear differential operators $D^{\epsilon}{_f}=zf_z-{\epsilon}{\bar{z}}f_{\bar{z}}({\mid}{\epsilon}{\mid}=1)$ and $F_{\lambda}(z)=(1-{\lambda)f+{\lambda}D^{\epsilon}{_f}(0{\leq}{\lambda}{\leq}1)$ when the coefficients of h and g satisfy harmonic Bieberbach coefficients conjecture conditions. Similar problems are also solved when the coefficients of h and g satisfy the corresponding necessary conditions of the harmonic convex function $f=h+{\bar{g}}$. All results are sharp. Some of the results are motivated by the work of Kalaj et al. [8].

ON A CERTAIN CLASS OF p-VALENT UNIFORMLY CONVEX FUNCTIONS USING DIFFERENTIAL OPERATOR

  • Lee, S.K.;Khairnar, S.M.;Rajas, S.M.
    • Korean Journal of Mathematics
    • /
    • v.19 no.1
    • /
    • pp.1-16
    • /
    • 2011
  • In this paper, using differential operator, we have introduce new class of p-valent uniformly convex functions in the unit disc U = {z : |z| < 1} and obtain the coefficient bounds, extreme bounds and radius of starlikeness for the functions belonging to this generalized class. Furthermore, partial sums $f_k(z)$ of functions $f(z)$ in the class $S^*({\lambda},{\alpha},{\beta})$ are considered. The various results obtained in this paper are sharp.

LOGHARMONIC MAPPINGS WITH TYPICALLY REAL ANALYTIC COMPONENTS

  • AbdulHadi, Zayid;Alarifi, Najla M.;Ali, Rosihan M.
    • Bulletin of the Korean Mathematical Society
    • /
    • v.55 no.6
    • /
    • pp.1783-1789
    • /
    • 2018
  • This paper treats the class of normalized logharmonic mappings $f(z)=zh(z){\overline{g(z)}}$ in the unit disk satisfying ${\varphi}(z)=zh(z)g(z)$ is analytically typically real. Every such mapping f admits an integral representation in terms of its second dilatation function and a function of positive real part with real coefficients. The radius of starlikeness and an upper estimate for arclength are obtained. Additionally, it is shown that f maps the unit disk into a domain symmetric with respect to the real axis when its second dilatation has real coefficients.

Radii of Starlikeness and Convexity for Analytic Functions with Fixed Second Coefficient Satisfying Certain Coefficient Inequalities

  • MENDIRATTA, RAJNI;NAGPAL, SUMIT;RAVICHANDRAN, V.
    • Kyungpook Mathematical Journal
    • /
    • v.55 no.2
    • /
    • pp.395-410
    • /
    • 2015
  • For functions $f(z)=z+a_2z^2+a_3z^3+{\cdots}$ with ${\mid}a_2{\mid}=2b$, $b{\geq}0$, sharp radii of starlikeness of order ${\alpha}(0{\leq}{\alpha}<1)$, convexity of order ${\alpha}(0{\leq}{\alpha}<1)$, parabolic starlikeness and uniform convexity are derived when ${\mid}a_n{\mid}{\leq}M/n^2$ or ${\mid}a_n{\mid}{\leq}Mn^2$ (M>0). Radii constants in other instances are also obtained.

Radius of Starlikeness for Analytic Functions with Fixed Second Coefficient

  • Ali, Rosihan M.;Kumar, Virendra;Ravichandran, V.;Kumar, Shanmugam Sivaprasad
    • Kyungpook Mathematical Journal
    • /
    • v.57 no.3
    • /
    • pp.473-492
    • /
    • 2017
  • Sharp radius constants for certain classes of normalized analytic functions with fixed second coefficient, to be in the classes of starlike functions of positive order, parabolic starlike functions, and Sokół-Stankiewicz starlike functions are obtained. Our results extend several earlier works.

On a Class of Univalent Functions Defined by Ruscheweyh Derivatives

  • SHAMS, S.;KULKARNI, S.R.;JAHANGIRI, JAY M.
    • Kyungpook Mathematical Journal
    • /
    • v.43 no.4
    • /
    • pp.579-585
    • /
    • 2003
  • A new class of univalent functions is defined by making use of the Ruscheweyh derivatives. We provide necessary and sufficient coefficient conditions, extreme points, integral representations, distortion bounds, and radius of starlikeness and convexity for this class.

  • PDF

ON CERTAIN CLASS OF ANALYTIC FUNCTIONS DEFINED BY CONVOLUTIONS

  • Kwon, Oh-Sang;Cho, Nak-Eun
    • East Asian mathematical journal
    • /
    • v.5 no.1
    • /
    • pp.57-67
    • /
    • 1989
  • We introduce a class $L_{\sigma}*({\alpha},{\beta},{\gamma})$ of functions defined by $f*S_{\sigma}(z)$ of f(z) and $S_{\sigma}(z)=z/(1-z)^{2(1-{\sigma})}$. The present paper is to determine extreme point, coefficient inequalities., distortion Theorem and radius of starlikeness and convexity for functions in $L_{\sigma}*({\alpha},{\beta},{\gamma})$. And we give fractional calculus.

  • PDF