A class of analytic functions defined by Ruscheweyh derivative

K. S. Padmanabhan; M. Jayamala

Annales Polonici Mathematici (1991)

  • Volume: 54, Issue: 2, page 167-178
  • ISSN: 0066-2216

Abstract

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The function f ( z ) = z p + ∑ k = 1 ∞ a p + k z p + k (p ∈ ℕ = 1,2,3,...) analytic in the unit disk E is said to be in the class K n , p ( h ) if ( D n + p f ) / ( D n + p - 1 f ) ≺ h , where D n + p - 1 f = ( z p ) / ( ( 1 - z ) p + n ) * f and h is convex univalent in E with h(0) = 1. We study the class K n , p ( h ) and investigate whether the inclusion relation K n + 1 , p ( h ) ⊆ K n , p ( h ) holds for p > 1. Some coefficient estimates for the class are also obtained. The class A n , p ( a , h ) of functions satisfying the condition a * ( D n + p f ) / ( D n + p - 1 f ) + ( 1 - a ) * ( D n + p + 1 f ) / ( D n + p f ) ≺ h is also studied.

How to cite

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K. S. Padmanabhan, and M. Jayamala. "A class of analytic functions defined by Ruscheweyh derivative." Annales Polonici Mathematici 54.2 (1991): 167-178. <http://eudml.org/doc/262247>.

@article{K1991,
abstract = {The function $f(z) = z^p + ∑_\{k=1\}^\{∞\} a_\{p+k\} z^\{p+k\}$ (p ∈ ℕ = 1,2,3,...) analytic in the unit disk E is said to be in the class $K_\{n,p\}(h)$ if ($D^\{n+p\}f)/(D^\{n+p-1\}f) ≺ h$, where $D^\{n+p-1\}f = (z^\{p\})/((1-z)^\{p+n\})*f$ and h is convex univalent in E with h(0) = 1. We study the class $K_\{n,p\}(h)$ and investigate whether the inclusion relation $K_\{n+1,p\}(h) ⊆ K_\{n,p\}(h)$ holds for p > 1. Some coefficient estimates for the class are also obtained. The class $A_\{n,p\}(a,h)$ of functions satisfying the condition $a*(D^\{n+p\}f)/(D^\{n+p-1\}f) + (1-a)*(D^\{n+p+1\}f)/(D^\{n+p\}f) ≺ h$ is also studied.},
author = {K. S. Padmanabhan, M. Jayamala},
journal = {Annales Polonici Mathematici},
keywords = {Hadamard product; coefficient estimates},
language = {eng},
number = {2},
pages = {167-178},
title = {A class of analytic functions defined by Ruscheweyh derivative},
url = {http://eudml.org/doc/262247},
volume = {54},
year = {1991},
}

TY - JOUR
AU - K. S. Padmanabhan
AU - M. Jayamala
TI - A class of analytic functions defined by Ruscheweyh derivative
JO - Annales Polonici Mathematici
PY - 1991
VL - 54
IS - 2
SP - 167
EP - 178
AB - The function $f(z) = z^p + ∑_{k=1}^{∞} a_{p+k} z^{p+k}$ (p ∈ ℕ = 1,2,3,...) analytic in the unit disk E is said to be in the class $K_{n,p}(h)$ if ($D^{n+p}f)/(D^{n+p-1}f) ≺ h$, where $D^{n+p-1}f = (z^{p})/((1-z)^{p+n})*f$ and h is convex univalent in E with h(0) = 1. We study the class $K_{n,p}(h)$ and investigate whether the inclusion relation $K_{n+1,p}(h) ⊆ K_{n,p}(h)$ holds for p > 1. Some coefficient estimates for the class are also obtained. The class $A_{n,p}(a,h)$ of functions satisfying the condition $a*(D^{n+p}f)/(D^{n+p-1}f) + (1-a)*(D^{n+p+1}f)/(D^{n+p}f) ≺ h$ is also studied.
LA - eng
KW - Hadamard product; coefficient estimates
UR - http://eudml.org/doc/262247
ER -

References

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  1. [1] P. Eenigenburg, S. S. Miller, P. T. Mocanu and M. O. Reade, On a Briot-Bouquet differential subordination, in: General Inequalities 3, Birkhäuser, Basel 1983, 339-348. 
  2. [2] R. M. Goel and N. S. Sohi, A new criterion for p-valent functions, Proc. Amer. Math. Soc. 78 (1980), 353-357. Zbl0444.30012
  3. [3] S. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc. 49 (1975), 109-115. Zbl0303.30006
  4. [4] T. Umezawa, Multivalently close-to-convex functions, Proc. Amer. Math. Soc. 8 (1957), 869-874. Zbl0080.28301

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