Weighted integral transforms involving convolution with some subclasses of analytic functions

Year 2021, Volume: 3 Issue: 1, 35 - 45, 02.06.2021

Syed Zakar Hussain Bukhari , Rehana Razzaq Imtiaz Ahmed

Abstract

Let A represent the class of analytic functions f de…ned in the open unit disk
U := fz 2 C : jzj < 1g such that f(0) = f0(0) 􀀀 1 = 0 and let P represent
the well-known class of Carathéodory functions p such that p(0) = 1 and
Re p(z) > 0; z 2 U: A functions p analytic in U such that p(0) = 1 belongs
to the class Pk for k  2; if and only if
p (z) =
1
2
Z2
0
1 + ze􀀀i
1 􀀀 ze􀀀i d() (z 2 U) ;
where () : 0    2 is a function of bounded variation satis…es the
conditions
R2
0
d() = 2 and
R2
0 jd()j  k: For some  2 R; & < 1; k  2
and
 0; let R
k(
; &) denote the class of functions f 2 A satisfying the
condition:
ei

(1 􀀀
)
f (z)
z
+
f0 (z) 􀀀 &

2 Pk (z 2 U) :
For f 2 R
k(
; &), we de…ne the integral transform=m (f) (z) =
R1
0
m(t) f(tz)
t dt;
where m is a non-negative real-valued weight function with
R1
0
m(t)dt = 1.
The main objective of this paper is to study conditions for invariance of
the integral transforms =m and other relevant properties in connection with
functions in the class R
k(
; &). Also by allowing parameters to vary, we may
encompass a large number of previously known results.
Key words and Phrases: Convolution; Gauss hypergeom

Keywords

Convolution, Gauss hypergeometric function, weighted integral transforms

References

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• [3] R. Balasubramanian, S. Ponnusamy and M. Vuorinen, On hypergeometric functions and function spaces, J. Comput. Appl. Math. 139(2)(2002), 299-322.