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Applications of the theory of differential subordination for functions with fixed initial coefficient to univalent functions

100%
Nagpal S., Ravichandran V.
Annales Polonici Mathematici
|
2012
|
tom 105
|
nr 3
225-238
EN
By using the theory of first-order differential subordination for functions with fixed initial coefficient, several well-known results for subclasses of univalent functions are improved by restricting the functions to have fixed second coefficient. The influence of the second coefficient of univalent functions becomes evident in the results obtained.
2
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Fully starlike and fully convex harmonic mappings of order α

100%
Nagpal S., Ravichandran V.
Annales Polonici Mathematici
|
2013
|
tom 108
|
nr 1
85-107
EN
The hereditary properties of convexity and starlikeness for conformal mappings do not generalize to univalent harmonic mappings. This failure leads to the notions of fully starlike and fully convex mappings. In this paper, properties of fully starlike mappings of order α and fully convex mappings of order α (0 ≤ α < 1) are studied; in particular, the bounds for the radius of full starlikeness of order α as well as the radius of full convexity of order α are determined for certain families of univalent harmonic mappings. Unlike the analytic case, convexity is not preserved under the convolution of univalent harmonic convex mappings. Given two univalent harmonic convex mappings f and g, the radius r₀ such that their harmonic convolution f*g is a univalent harmonic convex mapping in |z| < r₀ is also investigated.
3
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Convex combination of analytic functions

81%
Cho N. E., Jain N. K., Ravichandran V.
Open Mathematics
|
2017
|
tom 15
|
nr 1
331-339
EN
Radii of convexity, starlikeness, lemniscate starlikeness and close-to-convexity are determined for the convex combination of the identity map and a normalized convex function F given by f(z) = α z+(1−α)F(z).
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