<abstract xmlns="http://www.w3.org/1999/xhtml">The purpose of this paper is to introduce a class of functions ℱλ, λ ∈ [0, 1], consisting of analytic functions f normalized by f(0) = f´(0) − 1 = 0 in the open unit disk U which satisfies the subordination condition that<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2019-0012_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mtext>z</mtext><msup><mi>f</mi><mo>′</mo></msup><mrow><mo>(</mo><mtext>z</mtext><mo>)</mo></mrow><mo>/</mo><mrow><mo>{</mo><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>λ</mi></mrow><mo>)</mo></mrow><mtext>f</mtext><mrow><mo>(</mo><mtext>z</mtext><mo>)</mo></mrow><mo>+</mo><mi>λ</mi><mtext>z</mtext></mrow><mo>}</mo></mrow><mo>≺</mo><mtext>q</mtext><mrow><mo>(</mo><mtext>z</mtext><mo>)</mo></mrow><mo>,</mo><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mtext>z</mtext><mo>∈</mo><mo>𝕌</mo><mo>,</mo></mrow></math><tex-math>$${\rm{z}}f'\left( {\rm{z}} \right)/\left\{ {\left( {1 - \lambda } \right){\rm{f}}\left( {\rm{z}} \right) + \lambda {\rm{z}}} \right\} \prec {\rm{q}}\left( {\rm{z}} \right),\,\,\,\,\,{\rm{z}} \in {\rm{\mathbb{U},}}$$</tex-math></alternatives></disp-formula>where <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2019-0012_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mtext>q</mtext><mrow><mo>(</mo><mtext>z</mtext><mo>)</mo></mrow><mo>=</mo><msqrt><mrow><mn>1</mn><mo>+</mo><msup><mrow><mtext>z</mtext></mrow><mn>2</mn></msup></mrow></msqrt><mo>+</mo><mtext>z</mtext></mrow></math><tex-math>${\rm{q}}\left( {\rm{z}} \right) = \sqrt {1 + {{\rm{z}}^{\rm{2}}}} + {\rm{z}}$</tex-math></alternatives></inline-formula>. Some basic properties (including the radius of convexity) are obtained for this class of functions.</abstract>

The purpose of this paper is to introduce a class of functions ℱλ, λ ∈ [0, 1], consisting of analytic functions f normalized by f(0) = f´(0) − 1 = 0 in the open unit disk U which satisfies the subordination condition thatzf′(z)/{(1-λ)f(z)+λz}≺q(z),     z∈𝕌,$${\rm{z}}f'\left( {\rm{z}} \right)/\left\{ {\left( {1 - \lambda } \right){\rm{f}}\left( {\rm{z}} \right) + \lambda {\rm{z}}} \right\} \prec {\rm{q}}\left( {\rm{z}} \right),\,\,\,\,\,{\rm{z}} \in {\rm{\mathbb{U},}}$$where q(z)=1+z2+z${\rm{q}}\left( {\rm{z}} \right) = \sqrt {1 + {{\rm{z}}^{\rm{2}}}} + {\rm{z}}$. Some basic properties (including the radius of convexity) are obtained for this class of functions.

On a class of analytic functions governed by subordination

M. P. University of Agriculture and Technology

Faculty of Mathematics and Natural Sciences

Acta Universitatis Sapientiae, Mathematica

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"On a class of analytic functions governed by subordination"}

The purpose of this paper is to introduce a class of functions ℱλ, λ ∈ [0, 1], consisting of analytic functions f normalized by f(0) = f´(0) − 1 = 0 in the...

On a class of analytic functions governed by subordination

Published Online: Aug 17, 2019

Page range: 144 - 155

Received: Feb 03, 2018

DOI: https://doi.org/10.2478/ausm-2019-0012

Keywords
analytic functions, convex functions, starlike functions, differential subordination, radius of starlikeness, radius of convexity

© 2019 Ravinder Krishna Raina et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

On a class of analytic functions governed by subordination

Published Online: Aug 17, 2019

Page range: 144 - 155

Received: Feb 03, 2018

DOI: https://doi.org/10.2478/ausm-2019-0012

Keywordsanalytic functions, convex functions, starlike functions, differential subordination, radius of starlikeness, radius of convexity

© 2019 Ravinder Krishna Raina et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
analytic functions, convex functions, starlike functions, differential subordination, radius of starlikeness, radius of convexity