# 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

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topK. 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

top- [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] 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] S. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc. 49 (1975), 109-115. Zbl0303.30006
- [4] T. Umezawa, Multivalently close-to-convex functions, Proc. Amer. Math. Soc. 8 (1957), 869-874. Zbl0080.28301

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