<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper, we study a class of univalent functions <italic>f</italic> as defined by making use of the generalized Ruscheweyh derivatives involving a general fractional derivative operator, satisfying<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gm-2020-0007_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mo>Re</mo><mrow><mo>{</mo><mrow><mfrac><mrow><mi>z</mi><mrow><mo>(</mo><mrow><msubsup><mstyle><mi>J</mi></mstyle><mn>1</mn><mrow><mi>λ</mi><mo>,</mo><mi>μ</mi></mrow></msubsup><mi>f</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>'</mo></mrow><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>γ</mi></mrow><mo>)</mo></mrow><msubsup><mstyle><mi>J</mi></mstyle><mn>1</mn><mrow><mi>λ</mi><mo>,</mo><mi>μ</mi></mrow></msubsup><mi>f</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>+</mo><mi>γ</mi><msup><mi>z</mi><mn>2</mn></msup><mrow><mo>(</mo><mrow><msubsup><mstyle><mi>J</mi></mstyle><mn>1</mn><mrow><mi>λ</mi><mo>,</mo><mi>μ</mi></mrow></msubsup><mi>f</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>'</mo><mo>'</mo></mrow></mfrac></mrow><mo>}</mo></mrow><mo>></mo><mi>β</mi><mo>.</mo></mrow></math><tex-math>{\mathop{\rm Re}\nolimits} \left\{{{{z\left({{\bf{J}}_1^{\lambda,\mu}f\left(z \right)} \right)'} \over {\left({1 - \gamma} \right){\bf{J}}_1^{\lambda,\mu}f\left(z \right) + \gamma {z^2}\left({{\bf{J}}_1^{\lambda,\mu}f\left(z \right)} \right)''}}} \right\} > \beta.</tex-math></alternatives></disp-formula>A necessary and sufficient condition for a function to be in the class <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gm-2020-0007_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>A</mi><mi>γ</mi><mrow><mi>λ</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>ν</mi></mrow></msubsup><mrow><mo>(</mo><mrow><mi>n</mi><mo>,</mo><mi>β</mi></mrow><mo>)</mo></mrow></mrow></math><tex-math>A_\gamma ^{\lambda,\mu,\nu}\left({n,\beta} \right)</tex-math></alternatives></inline-formula> is obtained. Also, our paper includes linear combination, integral operators and we introduce the subclass <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gm-2020-0007_eq_003.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>A</mi><mrow><mi>γ</mi><mo>,</mo><msub><mi>c</mi><mi>m</mi></msub></mrow><mrow><mi>λ</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>ν</mi></mrow></msubsup><mrow><mo>(</mo><mrow><mn>1</mn><mo>,</mo><mi>β</mi></mrow><mo>)</mo></mrow></mrow></math><tex-math>A_{\gamma,{c_m}}^{\lambda,\mu,\nu}\left({1,\beta} \right)</tex-math></alternatives></inline-formula> consisting of functions with negative and fixed finitely many coefficients. We study some interesting properties of <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gm-2020-0007_eq_004.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>A</mi><mrow><mi>γ</mi><mo>,</mo><msub><mi>c</mi><mi>m</mi></msub></mrow><mrow><mi>λ</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>ν</mi></mrow></msubsup><mrow><mo>(</mo><mrow><mn>1</mn><mo>,</mo><mi>β</mi></mrow><mo>)</mo></mrow></mrow></math><tex-math>A_{\gamma,{c_m}}^{\lambda,\mu,\nu}\left({1,\beta} \right)</tex-math></alternatives></inline-formula>.</abstract>

In this paper, we study a class of univalent functions f as defined by making use of the generalized Ruscheweyh derivatives involving a general fractional derivative operator, satisfyingRe{z(J1λ,μf(z))'(1-γ)J1λ,μf(z)+γz2(J1λ,μf(z))''}>β.{\mathop{\rm Re}\nolimits} \left\{{{{z\left({{\bf{J}}_1^{\lambda,\mu}f\left(z \right)} \right)'} \over {\left({1 - \gamma} \right){\bf{J}}_1^{\lambda,\mu}f\left(z \right) + \gamma {z^2}\left({{\bf{J}}_1^{\lambda,\mu}f\left(z \right)} \right)''}}} \right\} > \beta.A necessary and sufficient condition for a function to be in the class Aγλ,μ,ν(n,β)A_\gamma ^{\lambda,\mu,\nu}\left({n,\beta} \right) is obtained. Also, our paper includes linear combination, integral operators and we introduce the subclass Aγ,cmλ,μ,ν(1,β)A_{\gamma,{c_m}}^{\lambda,\mu,\nu}\left({1,\beta} \right) consisting of functions with negative and fixed finitely many coefficients. We study some interesting properties of Aγ,cmλ,μ,ν(1,β)A_{\gamma,{c_m}}^{\lambda,\mu,\nu}\left({1,\beta} \right).

Some applications of generalized Ruscheweyh derivatives involving a general fractional derivative operator to a class of analytic functions with negative coefficients II

College of Science, Department of Mathematics

General Mathematics

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

In this paper, we study a class of univalent functions f as defined by making use of the generalized Ruscheweyh derivatives involving a general fractional...

Some applications of generalized Ruscheweyh derivatives involving a general fractional derivative operator to a class of analytic functions with negative coefficients II

Publié en ligne: 31 juil. 2020

Pages: 85 - 103

Reçu: 08 févr. 2010

Accepté: 12 juil. 2010

DOI: https://doi.org/10.2478/gm-2020-0007

Mots clés
Generalized Ruscheweyh derivatives, Fractional derivative, Integral operators, Linear combination, Radii of starlikeness, Extreme points, Weighted mean and Arithmetic mean

© 2020 Waggas Galib Atshan et al., published by Sciendo

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

Some applications of generalized Ruscheweyh derivatives involving a general fractional derivative operator to a class of analytic functions with negative coefficients II

Publié en ligne: 31 juil. 2020

Pages: 85 - 103

Reçu: 08 févr. 2010

Accepté: 12 juil. 2010

DOI: https://doi.org/10.2478/gm-2020-0007

Mots clésGeneralized Ruscheweyh derivatives, Fractional derivative, Integral operators, Linear combination, Radii of starlikeness, Extreme points, Weighted mean and Arithmetic mean

© 2020 Waggas Galib Atshan et al., published by Sciendo

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

Mots clés
Generalized Ruscheweyh derivatives, Fractional derivative, Integral operators, Linear combination, Radii of starlikeness, Extreme points, Weighted mean and Arithmetic mean