Korean J. Math. Vol. 24 No. 3 (2016) pp.409-445
DOI: https://doi.org/10.11568/kjm.2016.24.3.409

Application of convolution theory on non-linear integral operators

Main Article Content

Satwanti Devi
A. Swaminathan

Abstract

The class Wβδ(α,γ) defined in the domain |z|<1 satisfying
Reeiϕ((1α+2γ)(f/z)δ+(α3γ+γ[(11/δ)(zf/f)+1/δ(1+zf/f)])(f/z)δ(zf/f)β)>0,
with the conditions α0, β<1, γ0, δ>0 and ϕR generalizes a particular case of the largest subclass of univalent functions, namely the class of Bazilevi\v c functions. Moreover, for
0<δ1(1ζ), 0ζ<1, the class Cδ(ζ) be the subclass of normalized analytic functions such that
Re(1/δ(1+zf/f)+(11/δ)(zf/f))>ζ,|z|<1.
In the present work, the sufficient conditions on λ(t) are investigated, so that the non-linear integral transform
Vλδ(f)(z)=(01λ(t)(f(tz)/t)δdt)1/δ,|z|<1,
carries the functions from Wβδ(α,γ) into Cδ(ζ). Several interesting applications are provided for special choices of λ(t). These results are useful in the attempt to generalize the two most important extremal problems in this direction using duality techniques and provide scope for further research.



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