Properties for Close-to-Convex and Quasi-Convex Functions Using <i>q</i>-Linear Operator

Properties for Close-to-Convex and Quasi-Convex Functions Using <i>q</i>-Linear Operator

In this work, we describe the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">q</mi></semantics></math></inline-formula>-analogue of a multiplier–Rusch...

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Bibliographic Details
Main Authors:	Ekram E. Ali, Rabha M. El-Ashwah, Abeer M. Albalahi, Wael W. Mohammed
Format:	Article
Language:	English
Published:	MDPI AG 2025-03-01
Series:	Mathematics
Subjects:	analytic function q-starlike functions q-convex functions q-close-to-convex functions q-analogue Catas operator q-analogue of Ruscheweyh operator
Online Access:	https://www.mdpi.com/2227-7390/13/6/900
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Summary:	In this work, we describe the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">q</mi></semantics></math></inline-formula>-analogue of a multiplier–Ruscheweyh operator of a specific family of linear operators <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="fraktur">I</mi><mrow><mi mathvariant="fraktur">q</mi><mo>,</mo><mi>ρ</mi></mrow><mi>s</mi></msubsup><mrow><mo>(</mo><mi>ν</mi><mo>,</mo><mi>τ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, and we obtain findings related to geometric function theory (GFT) by utilizing approaches established through subordination and knowledge of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">q</mi></semantics></math></inline-formula>-calculus operators. By using this operator, we develop generalized classes of quasi-convex and close-to-convex functions in this paper. Additionally, the classes <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="fraktur">K</mi><mrow><mi mathvariant="fraktur">q</mi><mo>,</mo><mi>ρ</mi></mrow><mi>s</mi></msubsup><mrow><mo>(</mo><mi>ν</mi><mo>,</mo><mi>τ</mi><mo>)</mo></mrow><mfenced open="(" close=")"><mi>φ</mi></mfenced></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="fraktur">Q</mi><mrow><mi mathvariant="fraktur">q</mi><mo>,</mo><mi>ρ</mi></mrow><mi>s</mi></msubsup><mrow><mo>(</mo><mi>ν</mi><mo>,</mo><mi>τ</mi><mo>)</mo></mrow><mfenced open="(" close=")"><mi>φ</mi></mfenced></mrow></semantics></math></inline-formula> are introduced. The invariance of these recently formed classes under the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">q</mi></semantics></math></inline-formula>-Bernardi integral operator is investigated, along with a number of intriguing inclusion relationships between them. Additionally, several unique situations and the beneficial outcomes of these studies are taken into account.
ISSN:	2227-7390