<abstract xmlns="http://www.w3.org/1999/xhtml">

Let 𝒜 be the class of normalized functions which are analytic in the open unit disc. Jakson <italic>q</italic>-derivative represented by convolution operator
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gm-2023-0011_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><msub><mrow><mi>D</mi></mrow><mi>q</mi></msub><mi>f</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mi>z</mi></mfrac><mrow><mo>{</mo> <mrow><mi>f</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>★</mo><mfrac><mi>z</mi><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>q</mi><mi>z</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>1</mn><mi>z</mi></mrow><mo>)</mo></mrow></mrow></mfrac></mrow> <mo>}</mo></mrow><mo>,</mo><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi>f</mi><mo>∈</mo><mi>𝒜</mi><mo>,</mo><mi> </mi><mi> </mi><mi> </mi><mi>q</mi><mo>∈</mo><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math>
<tex-math>{D_q}f\left( z \right) = {1 \over z}\left\{ {f\left( z \right)\star{z \over {\left( {1 - qz} \right)\left( {1z} \right)}}} \right\},\,\,\,\,f \in \mathcal{A},\,\,\,q \in \left( {0,1} \right)</tex-math>
</alternatives>
</disp-formula>
is used to introduce a unified class <italic>Mq</italic>(<italic>α, β, γ</italic>), <italic>α</italic> ≥ 0, <italic>β, γ</italic> ∈ [0, 1) and its various mapping properties are studied. For <italic>q</italic> → 1−1, <italic>β</italic> = <italic>γ</italic> = 0 and <italic>α</italic> = 1, this class reduces to the class <italic>K</italic> of close-to-convex univalent functions. A number of interesting results such as <italic>q</italic>-Bernardi integral operator and inclusion relations are included as part of this study. Applications are also pointed out as consequences of the main results.
</abstract>

Let 𝒜 be the class of normalized functions which are analytic in the open unit disc. Jakson q-derivative represented by convolution operator



Dqf(z)=1z{ f(z)★z(1-qz)(1z) },    f∈𝒜,   q∈(0,1)
{D_q}f\left( z \right) = {1 \over z}\left\{ {f\left( z \right)\star{z \over {\left( {1 - qz} \right)\left( {1z} \right)}}} \right\},\,\,\,\,f \in \mathcal{A},\,\,\,q \in \left( {0,1} \right)


is used to introduce a unified class Mq(α, β, γ), α ≥ 0, β, γ ∈ [0, 1) and its various mapping properties are studied. For q → 1−1, β = γ = 0 and α = 1, this class reduces to the class K of close-to-convex univalent functions. A number of interesting results such as q-Bernardi integral operator and inclusion relations are included as part of this study. Applications are also pointed out as consequences of the main results.

On analytic functions related with <italic>q</italic>-generalized bounded Mocanu variation

General Mathematics

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

On analytic functions related with -generalized bounded Mocanu variation

Let 𝒜 be the class of normalized functions which are analytic in the open unit disc. Jakson q-derivative represented by convolution operator



Dqf(z)=1z{...

On analytic functions related with q-generalized bounded Mocanu variation

Published Online: Mar 07, 2024

Page range: 39 - 50

Received: Apr 15, 2023

Accepted: Nov 12, 2023

DOI: https://doi.org/10.2478/gm-2023-0011

Keywords
Analytic functions, -calculus, Close-to-convex, -derivative, Mocanu variation

© 2023 Khalida Inayat Noor et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

On analytic functions related with q-generalized bounded Mocanu variation

Published Online: Mar 07, 2024

Page range: 39 - 50

Received: Apr 15, 2023

Accepted: Nov 12, 2023

DOI: https://doi.org/10.2478/gm-2023-0011

KeywordsAnalytic functions, -calculus, Close-to-convex, -derivative, Mocanu variation

© 2023 Khalida Inayat Noor et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
Analytic functions, -calculus, Close-to-convex, -derivative, Mocanu variation