<abstract xmlns="http://www.w3.org/1999/xhtml"><p>In this paper, we find the necessary and sufficient conditions, inclusion relations for Poisson distribution series <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2019-0007_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>𝒦</mi><mrow><mo>(</mo><mrow><mtext>m</mtext><mo>,</mo><mtext>z</mtext></mrow><mo>)</mo></mrow><mo>=</mo><mtext>z</mtext><mo>+</mo><munderover><mo>∑</mo><mrow><mtext>n</mtext><mo>=</mo><mn>2</mn></mrow><mo>∞</mo></munderover><mrow><mfrac><mrow><msup><mrow><mtext>m</mtext></mrow><mrow><mtext>n</mtext><mo>-</mo><mn>1</mn></mrow></msup></mrow><mrow><mrow><mo>(</mo><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>!</mo></mrow></mfrac><msup><mrow><mtext>e</mtext></mrow><mrow><mo>-</mo><mtext>m</mtext></mrow></msup><msup><mrow><mtext>z</mtext></mrow><mtext>n</mtext></msup></mrow></mrow></math><tex-math>$\mathcal{K}\left( {{\rm{m, z}}} \right) = {\rm{z + }}\sum\limits_{{\rm{n}} = 2}^\infty {{{{{\rm{m}}^{{\rm{n}} - 1}}} \over {\left( {n - 1} \right)!}}{{\rm{e}}^{ - {\rm{m}}}}{{\rm{z}}^{\rm{n}}}} $</tex-math></alternatives></inline-formula> to be in the subclasses 𝒮(k, λ) and 𝒞(k, λ) of analytic functions with negative coefficients. Further, we obtain necessary and sufficient conditions for the integral operator <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2019-0007_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mtext>𝒢</mtext><mrow><mo>(</mo><mrow><mtext>m</mtext><mo>,</mo><mtext>z</mtext></mrow><mo>)</mo></mrow><mo>=</mo><mrow><msubsup><mo>∫</mo><mn>0</mn><mtext>z</mtext></msubsup><mrow><mfrac><mrow><mtext>ℱ</mtext><mrow><mo>(</mo><mrow><mtext>m</mtext><mo>,</mo><mtext>t</mtext></mrow><mo>)</mo></mrow></mrow><mtext>t</mtext></mfrac></mrow></mrow><mtext>dt</mtext></mrow></math><tex-math>${\rm{\mathcal{G}}}\left( {{\rm{m}},{\rm{z}}} \right) = \int_0^{\rm{z}} {{{{\rm{\mathcal{F}}}\left( {{\rm{m}},{\rm{t}}} \right)} \over {\rm{t}}}} {\rm{dt}}$</tex-math></alternatives></inline-formula> to be in the above classes.</p></abstract>

In this paper, we find the necessary and sufficient conditions, inclusion relations for Poisson distribution series 𝒦(m,z)=z+∑n=2∞mn-1(n-1)!e-mzn$\mathcal{K}\left( {{\rm{m, z}}} \right) = {\rm{z + }}\sum\limits_{{\rm{n}} = 2}^\infty {{{{{\rm{m}}^{{\rm{n}} - 1}}} \over {\left( {n - 1} \right)!}}{{\rm{e}}^{ - {\rm{m}}}}{{\rm{z}}^{\rm{n}}}} $ to be in the subclasses 𝒮(k, λ) and 𝒞(k, λ) of analytic functions with negative coefficients. Further, we obtain necessary and sufficient conditions for the integral operator 𝒢(m,z)=∫0zℱ(m,t)tdt${\rm{\mathcal{G}}}\left( {{\rm{m}},{\rm{z}}} \right) = \int_0^{\rm{z}} {{{{\rm{\mathcal{F}}}\left( {{\rm{m}},{\rm{t}}} \right)} \over {\rm{t}}}} {\rm{dt}}$ to be in the above classes.

On certain subclasses of analytic functions associated with Poisson distribution series

Acta Universitatis Sapientiae, Mathematica

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

In this paper, we find the necessary and sufficient conditions, inclusion relations for Poisson distribution series...

On certain subclasses of analytic functions associated with Poisson distribution series

Published Online: Aug 17, 2019

Page range: 78 - 86

Received: Aug 25, 2018

DOI: https://doi.org/10.2478/ausm-2019-0007

Keywords
analytic functions, Poisson distribution series

© 2019 B. A. Frasin, published by Sciendo

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

On certain subclasses of analytic functions associated with Poisson distribution series

Published Online: Aug 17, 2019

Page range: 78 - 86

Received: Aug 25, 2018

DOI: https://doi.org/10.2478/ausm-2019-0007

Keywordsanalytic functions, Poisson distribution series

© 2019 B. A. Frasin, published by Sciendo

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

Keywords
analytic functions, Poisson distribution series