Application of the <i>q</i>-derivative operator to a specialized class of harmonic functions exhibiting positive real part

Khadeejah Rasheed Alhindi; Khadeejah Rasheed Alhindi

doi:10.3934/math.2025090

AIMS Mathematics

2025, Volume 10, Issue 1: 1935-1944. doi: 10.3934/math.2025090

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Research article

Application of the q-derivative operator to a specialized class of harmonic functions exhibiting positive real part

Khadeejah Rasheed Alhindi ^,

Department of Computer Science, College of Computer Science and Information Technology, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam 31441, Saudi Arabia

Received: 01 December 2024 Revised: 13 January 2025 Accepted: 16 January 2025 Published: 06 February 2025
MSC : 30C45

This paper introduces a new subclass of harmonic functions with a positive real part, denoted by $ HP_q(\beta) $, where $ 0 \leq \beta < 1 $ and $ 0 < q < 1 $. A sufficient coefficient condition is established for functions within this class, which is also necessary when dealing with negative coefficients. In addition, the growth theorem is derived, and the extreme points associated with this subclass are also identified. Finally, the $ q $-integral operator for harmonic functions of the form $ f = h + g $ with a positive real part is presented.
- q-derivative operator,
- q-integral operator,
- q-calculus,
- quantum calculus,
- harmonic functions,
- positive real part,
- growth theorem,
- extreme points
Citation: Khadeejah Rasheed Alhindi. Application of the q-derivative operator to a specialized class of harmonic functions exhibiting positive real part[J]. AIMS Mathematics, 2025, 10(1): 1935-1944. doi: 10.3934/math.2025090

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Abstract

This paper introduces a new subclass of harmonic functions with a positive real part, denoted by $ HP_q(\beta) $, where $ 0 \leq \beta < 1 $ and $ 0 < q < 1 $. A sufficient coefficient condition is established for functions within this class, which is also necessary when dealing with negative coefficients. In addition, the growth theorem is derived, and the extreme points associated with this subclass are also identified. Finally, the $ q $-integral operator for harmonic functions of the form $ f = h + g $ with a positive real part is presented.

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