Geometric behavior of a class of algebraic differential equations in a complex domain using a majorization concept

Rabha W. Ibrahim; Dumitru Baleanu; Rabha W. Ibrahim; Dumitru Baleanu

doi:10.3934/math.2021049

AIMS Mathematics

2021, Volume 6, Issue 1: 806-820. doi: 10.3934/math.2021049

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Research article

Geometric behavior of a class of algebraic differential equations in a complex domain using a majorization concept

Rabha W. Ibrahim ^{1,2
,
,},
Dumitru Baleanu ^3,4,5

1.
Informetrics Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam
2.
Faculty of Mathematics & Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
3.
Department of Mathematics, Cankaya University, 06530 Balgat, Ankara, Turkey
4.
Institute of Space Sciences, R76900 Magurele-Bucharest, Romania
5.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan

Received: 24 August 2020 Accepted: 21 October 2020 Published: 02 November 2020
MSC : 30C55, 30C45

In this paper, a type of complex algebraic differential equations (CADEs) is considered formulating by $ \alpha [\varphi(z) \varphi" (z) +(\varphi' (z))^2]+ a_m \varphi^m(z)+a_{m-1} \varphi^{m-1}(z)+...+ a_1 \varphi(z)+ a_0 = 0. $ The conformal analysis (angle-preserving) of the CADEs is investigated. We present sufficient conditions to obtain analytic solutions of the CADEs. We show that these solutions are subordinated to analytic convex functions in terms of $e^z.$ Moreover, we investigate the connection estimates (coefficient bounds) of CADEs by employing the majorization method. We achieve that the coefficients bound are optimized by Bernoulli numbers.
- analytic function,
- subordination and superordination,
- univalent function,
- open unit disk,
- algebraic differential equations,
- majorization method
Citation: Rabha W. Ibrahim, Dumitru Baleanu. Geometric behavior of a class of algebraic differential equations in a complex domain using a majorization concept[J]. AIMS Mathematics, 2021, 6(1): 806-820. doi: 10.3934/math.2021049

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Abstract

In this paper, a type of complex algebraic differential equations (CADEs) is considered formulating by $ \alpha [\varphi(z) \varphi" (z) +(\varphi' (z))^2]+ a_m \varphi^m(z)+a_{m-1} \varphi^{m-1}(z)+...+ a_1 \varphi(z)+ a_0 = 0. $ The conformal analysis (angle-preserving) of the CADEs is investigated. We present sufficient conditions to obtain analytic solutions of the CADEs. We show that these solutions are subordinated to analytic convex functions in terms of $e^z.$ Moreover, we investigate the connection estimates (coefficient bounds) of CADEs by employing the majorization method. We achieve that the coefficients bound are optimized by Bernoulli numbers.

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