<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

Online veröffentlicht: 17. Aug. 2019

Seitenbereich: 78 - 86

Eingereicht: 25. Aug. 2018

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

Schlüsselwörter
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

Online veröffentlicht: 17. Aug. 2019

Seitenbereich: 78 - 86

Eingereicht: 25. Aug. 2018

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

Schlüsselwörteranalytic 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.

Schlüsselwörter
analytic functions, Poisson distribution series