Bi-univalent functions subordinated to a three leaf function induced by multiplicative calculus

G. Murugusundaramoorthy; K. Vijaya; K. R. Karthikeyan; Sheza M. El-Deeb; Jong-Suk Ro; G. Murugusundaramoorthy; K. Vijaya; K. R. Karthikeyan; Sheza M. El-Deeb; Jong-Suk Ro

doi:10.3934/math.20241313

AIMS Mathematics

2024, Volume 9, Issue 10: 26983-26999. doi: 10.3934/math.20241313

Previous Article Next Article

Research article

Bi-univalent functions subordinated to a three leaf function induced by multiplicative calculus

1.
Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology (VIT), Vellore 632014, India
2.
Department of Applied Mathematics and Science, College of Engineering, National University of Science & Technology, P. O. Box 2322, CPO Seeb 111, Al Hail, Muscat, Oman
3.
Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia
4.
School of Electrical and Electronics Engineering, Chung-Ang University, Dongjak-gu, Seoul 06974, Republic of Korea
5.
Department of Intelligent Energy and Industry, Chung-Ang University, Dongjak-gu, Seoul 06974, Republic of Korea

Received: 29 June 2024 Revised: 29 August 2024 Accepted: 09 September 2024 Published: 18 September 2024
MSC : 30C45

Our aim was to develop a new class of bi starlike functions by utilizing the concept of subordination, driven by the idea of multiplicative calculus, specifically multiplicative derivatives. Several restrictions were imposed, which were indeed strict constraints, because we have tried to work within the current framework or the design of analytic functions. To make the study more versatile, we redefined our new class of function with Miller-Ross Poisson distribution (MRPD), in order to increase the study's adaptability. We derived the first coefficient estimates and Fekete-Szegő inequalities for functions in this new class. To demonstrate the characteristics, we have provided a few examples.

Keywords:

Citation: G. Murugusundaramoorthy, K. Vijaya, K. R. Karthikeyan, Sheza M. El-Deeb, Jong-Suk Ro. Bi-univalent functions subordinated to a three leaf function induced by multiplicative calculus[J]. AIMS Mathematics, 2024, 9(10): 26983-26999. doi: 10.3934/math.20241313

Related Papers:

[1]	Min Li, Ke Chen, Yunqing Bai, Jihong Pei . Skeleton action recognition via graph convolutional network with self-attention module. Electronic Research Archive, 2024, 32(4): 2848-2864. doi: 10.3934/era.2024129
[2]	Wangwei Zhang, Menghao Dai, Bin Zhou, Changhai Wang . MCADFusion: a novel multi-scale convolutional attention decomposition method for enhanced infrared and visible light image fusion. Electronic Research Archive, 2024, 32(8): 5067-5089. doi: 10.3934/era.2024233
[3]	Bingsheng Li, Na Li, Jianmin Ren, Xupeng Guo, Chao Liu, Hao Wang, Qingwu Li . Enhanced spectral attention and adaptive spatial learning guided network for hyperspectral and LiDAR classification. Electronic Research Archive, 2024, 32(7): 4218-4236. doi: 10.3934/era.2024190
[4]	Hui Xu, Jun Kong, Mengyao Liang, Hui Sun, Miao Qi . Video behavior recognition based on actional-structural graph convolution and temporal extension module. Electronic Research Archive, 2022, 30(11): 4157-4177. doi: 10.3934/era.2022210
[5]	Huimin Qu, Haiyan Xie, Qianying Wang . Multi-convolutional neural network brain image denoising study based on feature distillation learning and dense residual attention. Electronic Research Archive, 2025, 33(3): 1231-1266. doi: 10.3934/era.2025055
[6]	Kun Zheng, Li Tian, Zichong Li, Hui Li, Junjie Zhang . Incorporating eyebrow and eye state information for facial expression recognition in mask-obscured scenes. Electronic Research Archive, 2024, 32(4): 2745-2771. doi: 10.3934/era.2024124
[7]	Bojian Chen, Wenbin Wu, Zhezhou Li, Tengfei Han, Zhuolei Chen, Weihao Zhang . Attention-guided cross-modal multiple feature aggregation network for RGB-D salient object detection. Electronic Research Archive, 2024, 32(1): 643-669. doi: 10.3934/era.2024031
[8]	Jiange Liu, Yu Chen, Xin Dai, Li Cao, Qingwu Li . MFCEN: A lightweight multi-scale feature cooperative enhancement network for single-image super-resolution. Electronic Research Archive, 2024, 32(10): 5783-5803. doi: 10.3934/era.2024267
[9]	Wangwei Zhang, Hao Sun, Bin Zhou . TBRAFusion: Infrared and visible image fusion based on two-branch residual attention Transformer. Electronic Research Archive, 2025, 33(1): 158-180. doi: 10.3934/era.2025009
[10]	Jingqian Xu, Ma Zhu, Baojun Qi, Jiangshan Li, Chunfang Yang . AENet: attention efficient network for cross-view image geo-localization. Electronic Research Archive, 2023, 31(7): 4119-4138. doi: 10.3934/era.2023210

Abstract

1. Introduction

The study on flow of fluids which are electrically conducting is known as magnetohydrodynamics (MHD). The magnetohydrodynamics have important applications in the polymer industry and engineering fields (Garnier ^[1]). Heat transfer caused by hydromagnetism was discussed by Chakrabarthi and Gupta ^[2]. Using an exponentially shrinking sheet, Nadeem et al. ^[3] investigated the MHD flow of a Casson fluid. Krishnendu Bhattacharyya ^[4] examined the effect of thermal radiation on MHD stagnation-point Flow over a Stretching Sheet.

Mixed convection magnetohydrodynamics flow is described by Ishak on a vertical and on a linearly stretching sheet ^[5,6,7]. Hayat et al. ^[10] examine a mixed convection flow within a stretched sheet of Casson nanofluid. Subhas Abel and Monayya Mareppa, examine magnetohydrodynamics flow on a vertical plate ^[11]. Shen et al. ^[12], examined a vertical stretching sheet which was non-linear. Ishikin Abu Bakar ^[13] investigates how boundary layer flow is affected by slip and convective boundary conditions over a stretching sheet. A vertical plate oscillates with the influence of slip on a free convection flow of a Casson fluid ^[14].

Nasir Uddin et al. ^[15] used a Runge-Kutta sixth-order integration method. Barik et al. ^[16] implicit finite distinction methodology of Crank Sir Harold George Nicolson sort Raman and Kumar ^[17] utilized an exact finite distinction theme of DuFort–Frankel. Mondal et al. ^[18] used a numerical theme over the whole vary of physical parameters. With the laplace transform method, we can determine the magnetohydrodynamic flow of a viscous fluid ^[19]. Thamizh Suganya et al. ^[20] obtained that the MHD for the free convective flow of fluid is based on coupled non-linear differential equations. In this study, the analytical approximation of concentration profiles in velocity, temperature and concentration using homotopy perturbation method (HPM).

2. Mathematical construction of the problem

The governing differential equations in dimensionless form ^[19] as follows:

$\begin{align} \frac{d^2 u}{dy^2} - Hu + Gr \theta + Gm \phi = 0, \end{align}$

(2.1)

$\begin{align} \frac{1}{F} \frac{d^2 \theta}{dy^2} = 0, \end{align}$

(2.2)

$\begin{align} \frac{1}{Sc} \frac{d^2 \phi}{dy^2} - Sr \frac{\partial^2 \phi}{\partial y^2} - \gamma \phi = 0. \end{align}$

(2.3)

The dimensionless boundary conditions given by:

$\begin{equation} u = 0, \ \theta = 1, \phi = 0 \ \ at \ \ y = 0 \end{equation}$

(2.4)

and

$\begin{equation} u = 0, \ \theta = 0, \phi = 0 \ \ at \ \ y \to \infty. \end{equation}$

(2.5)

3. Solutions of steady-state concentration profile using the Homotopy Perturbation Method

He ^[21,22] established the homotopy perturbation method, which waives the requirement of small parameters. Many researchers have used HPM to obtain approximate analytical solutions for many non-linear engineering dynamical systems ^[23,24]. The basic concept of the HPM as follows:

$\begin{align} \frac{d^2 u}{dy^2} - Hu + Gr \theta + Gm \phi = 0 \end{align}$

(3.1)

$\begin{align} \frac{1}{F} \frac{d^2 \theta}{dy^2} = 0 \end{align}$

(3.2)

$\begin{align} \frac{1}{Sc} \frac{d^2 \phi}{dy^2} - Sr \frac{\partial^2 \phi}{\partial y^2} - \gamma \phi = 0 \end{align}$

(3.3)

with initial and boundary conditions given by:

$\begin{align} y = 0 \ \ &at \ u = 0, \ \theta = 1, C = 1 \\ y \to \infty \ \ &as \ u = 0, \ \theta = 1, \phi = 0. \end{align}$

(3.4)

Homotopy for the above Eqs (3.1) to (3.4) can be constructed as follows:

$\begin{align} (1 - p) \left[\frac{d^2 u}{dy^2} - Hu + Gr \theta + Gm \phi\right] + p \left[\frac{d^2 u}{dy^2} - Hu + Gr \theta + Gm \phi\right] = 0 \end{align}$

(3.5)

$\begin{align} (1 - p) \left[\frac{1}{F} \frac{d^2 \theta}{dy^2} - \theta\right] + p \left[\frac{1}{F} \frac{d^2 \theta}{dy^2} - \theta + \theta\right] = 0 \end{align}$

(3.6)

$\begin{align} (1 - p) \left[\frac{1}{Sc} \frac{d^2 \phi}{dy^2} - \gamma \phi \right] + p \left[\frac{1}{Sc} \frac{d^2 \phi}{dy^2} - Sr \frac{\partial^2 \theta}{\partial y^2} - \gamma \phi \right] = 0 \end{align}$

(3.7)

The approximate solution of the Eqs (3.5) to (3.7) are

$\begin{align} u & = u_0 + pu_1 + p^2 u_2 + p^3 u_3 + ... \end{align}$

(3.8)

$\begin{align} \theta & = \theta_0 + p \theta_1 + p^2 \theta_2 + p^3 \theta_3 + ... \end{align}$

(3.9)

$\begin{align} \phi & = \phi_0 + p \phi_1 + p^2 \phi_2 + p^3 \phi_3 + ... \end{align}$

(3.10)

Substitution Eqs (3.5) to (3.7) in Eqs (3.8) to (3.10) respectively. We obtain the following equations

$\begin{align} (1 - p) &\left[\frac{d^2 (u_0 + pu_1 + p^2 u_2 + p^3 u_3 + ...)}{dy^2} - H(u_0 + pu_1 + p^2 u_2 + p^3 u_3 + ...) \right. \\ &\left.+ Gr (\theta_0 + p \theta_1 + p^2 \theta_2 + p^3 \theta_3 + ...) + Gm (\phi_0 + p \phi_1 + p^2 \phi_2 + p^3 \phi_3 + ...)\right] \\ &+ p\left[\frac{d^2 (u_0 + pu_1 + p^2 u_2 + p^3 u_3 + ...)}{dy^2} - H(u_0 + pu_1 + p^2 u_2 + p^3 u_3 + ...)\right. \\ &\left.+ Gr (\theta_0 + p \theta_1 + p^2 \theta_2 + p^3 \theta_3 + ...) + Gm (\phi_0 + p \phi_1 + p^2 \phi_2 + p^3 \phi_3 + ...)\right] = 0 \end{align}$

(3.11)

and

$\begin{align} (1 - p) &\left[\frac{1}{F} \frac{d^2 (\theta_0 + p \theta_1 + p^2 \theta_2 + p^3 \theta_3 + ...)}{dy^2} - (\theta_0 + p \theta_1 + p^2 \theta_2 + p^3 \theta_3 + ...)\right] \\ &+ p \left[\frac{1}{F} \frac{d^2 (\theta_0 + p \theta_1 + p^2 \theta_2 + p^3 \theta_3 + ...)}{dy^2} - (\theta_0 + p \theta_1 + p^2 \theta_2 + p^3 \theta_3 + ...) \right. \\ &\left.+ (\theta_0 + p \theta_1 + p^2 \theta_2 + p^3 \theta_3 + ...)\right] = 0, \end{align}$

(3.12)

and

$\begin{align} (1 - p) &\left[\frac{1}{Sc} \frac{d^2 (\phi_0 + p \phi_1 + p^2 \phi_2 + p^3 \phi_3 + ...)}{dy^2} - \gamma (\phi_0 + p \phi_1 + p^2 \phi_2 + p^3 \phi_3 + ...)\right] \\ &+ p \left[\frac{1}{Sc} \frac{d^2 (\phi_0 + p \phi_1 + p^2 \phi_2 + p^3 \phi_3 + ...)}{dy^2} - Sr \frac{\partial^2 (\theta_0 + p \theta_1 + p^2 \theta_2 + p^3 \theta_3 + ...)}{\partial y^2} \right. \\ &\left.- \gamma (\phi_0 + p \phi_1 + p^2 \phi_2 + p^3 \phi_3 + ...) \right] = 0. \end{align}$

(3.13)

Equating the coefficient of $p$ on both sides, we get the following equations

$\begin{align} P^0 &: \frac{d^2 u_0}{dy^2} - Hu_0 + Gr \theta_0 + Gm \phi_0 = 0; \end{align}$

(3.14)

$\begin{align} p^0 &: \frac{1}{F} \frac{d^2 \theta_0}{dy^2} - \theta_0 = 0; \end{align}$

(3.15)

$\begin{align} P^1 &: \frac{1}{F} \frac{d^2 \theta_0}{dy^2} - \theta_0 + \theta_1 = 0; \end{align}$

(3.16)

$\begin{align} P^1 &: \frac{1}{Sc} \frac{d^2 \phi_0}{dy^2} - \gamma \phi_0 = 0; \end{align}$

(3.17)

$\begin{align} P^1 &: \frac{1}{Sc} \frac{d^2 \phi_0}{dy^2} - Sr \frac{\partial^2 \theta_0}{\partial y^2} - \gamma \phi_0 = 0. \end{align}$

(3.18)

The boundary conditions are

$\begin{align} u_0 = 0, \ \theta_0 = 1, \phi_0 = 1 \ \ at \ \ y = 0 \\ u_0 = 0, \ \theta_0 = 1, \phi_0 = 1 \ \ at \ \ y \to \infty. \end{align}$

(3.19)

and

$\begin{align} u_1 = 0, \ \theta_1 = 0, \phi_1 = 0 \ \ at \ \ y = 0 \\ u_1 = 0, \ \theta_1 = 0, \phi_1 = 0 \ \ at \ \ y \to \infty. \end{align}$

(3.20)

Solving the Eqs (3.9)–(3.14), we obtain

$\begin{align} u_0(t) & = \frac{Gr}{\sqrt{\frac{1}{F}} + H} \left[e^{-y \sqrt{\frac{1}{F}}} - e^{-y \sqrt{H}}\right] + \frac{Gm}{\sqrt{\gamma Sc + H}} \left[e^{-y \sqrt{\gamma Sc}} - e^{-y \sqrt{H}}\right]; \end{align}$

(3.21)

$\begin{align} \theta_0(y) & = e^{-y \sqrt{\frac{1}{F}}}; \end{align}$

(3.22)

$\begin{align} \phi_0(y) & = e^{-y \sqrt{\gamma Sc}}; \end{align}$

(3.23)

$\begin{align} \phi_1(y) & = \frac{Sc Sr}{\sqrt{F} + F \gamma Sc} \left[e^{-y \sqrt{\frac{1}{F}}} - e^{-y \sqrt{\gamma Sc}}\right]. \end{align}$

(3.24)

Considering the iteration, we get,

$\begin{align} u(t) & = \frac{Gr}{\sqrt{\frac{1}{F}} + H} \left[e^{-y \sqrt{\frac{1}{F}}} - e^{-y \sqrt{H}}\right] + \frac{Gm}{\sqrt{\gamma Sc + H}} \left[e^{-y \sqrt{\gamma Sc}} - e^{-y \sqrt{H}}\right]; \end{align}$

(3.25)

$\begin{align} \theta(y) & = e^{-y \sqrt{F}}; \end{align}$

(3.26)

$\begin{align} \phi(y) & = e^{-y \sqrt{\gamma Sc}} + \frac{Sc Sr}{\sqrt{F} + F \gamma Sc} \left[e^{-y \sqrt{\frac{1}{F}}} - e^{-y \sqrt{\gamma Sc}}\right]. \end{align}$

(3.27)

4. Approximate analytical solutions for the skiin friction, Nusselt and Sherwood numbers

From the Eqs (3.25)–(3.27), we obtain

$\begin{array}{l} C_f = - \left(\frac{\partial u}{\partial y}\right)_{y = 0} \\ = \frac{\left(Gm \sqrt{\gamma Sc} - Gm \sqrt{H} + Gr(\gamma Sc + H) \sqrt{1 + \frac{R}{Pr}} + Gm H \sqrt{\gamma Sc} + (-Gm - Gr) H^{\frac{3}{2}} - \gamma Gr \sqrt{H} Sc\right)}{\left(\sqrt{1 + \frac{R}{Pr} + H} (\gamma Sc + H)\right)}; \end{array}$

(4.1)

$\begin{align} N_u & = - \left(\frac{\partial \theta}{\partial y}\right)_{y = 0} = \sqrt{\frac{1 + R}{Pr}}; \end{align}$

(4.2)

$\begin{array}{l} S_h = - \left(\frac{\partial \phi}{\partial y}\right)_{y = 0} \\= \frac{Sc Sr (1 + R) \sqrt{1 + \frac{R}{Pr}} - \sqrt{\gamma Sc} \left((-R - 1) \sqrt{1 + \frac{R}{Pr}} + ((1 + R)Sc - Pr \gamma) Sc \right)}{(1 + R) \sqrt{1 + \frac{R}{Pr}} Pr \gamma Sc} \end{array}$

(4.3)

5. Results and discussion

The combined impacts of transient MHD free convective flows of an incompressible viscous fluid through a vertical plate moving with uniform motion and immersed in a porous media are examined using an exact approach. The approximate analytical expressions for the velocity $u$ , temperature $\theta$ , and concentration profile $\phi$ are solved by using the homotopy perturbation method for fixed values of parameters is graphically presented.

Velocity takes time at first, and for high values of y, it takes longer, and the velocity approaches zero as time increases. The velocity of fluid rises with Gr increasing, as exposed in Figure 1.

Figure 1. An illustration of velocity profiles for different values of Gr.

[1]	P. L. Duren, Univalent functions, New York: Springer, 1983.
[2]	C. Pommerenke, G. Jensen, Univalent functions, Göttingen: Vandenhoeck und Ruprecht, 1975.
[3]	A. W. Goodman, Univalent functions, Volume I, Tampa: Mariner Pub. Co., 1983.
[4]	D. A. Brannan, T. S. Taha, On some classes of bi-univalent functions, Stud. U. Babes-Bol. Mat., 31 (1986), 70–77.
[5]	T. S. Taha, Topics in univalent function theory, PhD Thesis, University of London, 1981.
[6]	D. A. Brannan, J. G. Clunie, Aspects of contemporary complex analysis, New York: Academic Press, 1980.
[7]	E. Deniz, Certain subclasses of bi-univalent functions satisfying subordinate conditions, Journal of Classical Analysis, 2 (2013), 49–60. https://doi.org/10.7153/jca-02-05 doi: 10.7153/jca-02-05
[8]	M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc., 18 (1967), 63–68. https://doi.org/10.2307/2035225 doi: 10.2307/2035225
[9]	G. Murugusundaramoorthy, K. Vijaya, T. Bulboacă, Initial coefficient bounds for bi-univalent functions related to Gregory coefficients, Mathematics, 11 (2023), 2857. https://doi.org/10.3390/math11132857 doi: 10.3390/math11132857
[10]	E. Netanyahu, The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in $\|z\| < 1$ , Arch. Rational Mech. Anal., 32 (1969), 100–112. https://doi.org/10.1007/BF00247676 doi: 10.1007/BF00247676
[11]	H. M. Srivastava, A. K. Mishra, P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett., 23 (2010), 1188–1192. https://doi.org/10.1016/j.aml.2010.05.009 doi: 10.1016/j.aml.2010.05.009
[12]	H. M. Srivastava, G. Murugusundaramoorthy, T. Bulboacă, The second Hankel determinant for subclasses of bi-univalent functions associated with a nephroid domain, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat., 116 (2022), 145. https://doi.org/10.1007/s13398-022-01286-6 doi: 10.1007/s13398-022-01286-6
[13]	K. Vijaya, G. Murugusundaramoorthy, Bi-starlike function of complex order involving Mathieu-type series associated with telephone numbers, Symmetry, 15 (2023), 638. https://doi.org/10.3390/sym15030638 doi: 10.3390/sym15030638
[14]	H. Tang, G.-T. Deng, S.-H. Li, Coefficient estimates for new subclasses of Ma-Minda bi-univalent functions, J. Inequal. Appl., 2013 (2013), 317. https://doi.org/10.1186/1029-242X-2013-317 doi: 10.1186/1029-242X-2013-317
[15]	W. C. Ma, D. Minda, A unified treatment of some special classes of univalent functions, In: Proceedings of the conference on complex analysis (Tianjin, 1992), Cambridge: Internat. Press, 1992,157–169.
[16]	S. Gandhi, Radius Estimates for three leaf function and convex combination of starlike functions, In: Mathematical analysis I: approximation theory, Singapore: Springer, 2020,173–184. https://doi.org/10.1007/978-981-15-1153-0_15
[17]	K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, New York: John Wiley and Sons, 1993.
[18]	A. Wiman, Über die Nullstellen der Funktionen $E_\alpha(x)$ , Acta Math., 29 (1905), 217–234. https://doi.org/10.1007/BF02403204 doi: 10.1007/BF02403204
[19]	S. S. Eker, S. Ece, Geometric properties of the Miller-Ross functions, Iran. J. Sci. Technol. Trans. Sci., 46 (2022), 631–636. https://doi.org/10.1007/s40995-022-01268-8 doi: 10.1007/s40995-022-01268-8
[20]	S. M. El-Deeb, T. Bulboacă, J. Dziok, Pascal distribution series connected with certain subclasses of univalent functions, Kyungpook Math. J., 59 (2019), 301–314. https://doi.org/10.5666/KMJ.2019.59.2.301 doi: 10.5666/KMJ.2019.59.2.301
[21]	W. Nazeer, Q. Mehmood, S. M. Kang, A. U. Haq, An application of Bionomial distribution series on certain analytic functions, J. Comput. Anal. Appl., 26 (2019), 11–17.
[22]	S. Porwal, M. Kumar, A unified study on starlike and convex functions associated with Poisson distribution series, Afr. Mat., 27 (2016), 1021–1027. https://doi.org/10.1007/s13370-016-0398-z doi: 10.1007/s13370-016-0398-z
[23]	A. K. Wanas, N. A. Al-Ziadi, Applications of beta negative binomial distribution series on holomorphic funxtions, Earthline Journal of Mathematical Sciences, 6 (2021), 271–292. https://doi.org/10.34198/ejms.6221.271292 doi: 10.34198/ejms.6221.271292
[24]	A. K. Wanas, J. A. Khuttar, Applications of Borel distribution series on analytic functions, Earthline Journal of Mathematical Sciences, 4 (2020), 71–82. https://doi.org/10.34198/ejms.4120.7182 doi: 10.34198/ejms.4120.7182
[25]	A. E. Bashirov, E. M. Kurpinar, A. Özyapıcı, Multiplicative calculus and its applications, J. Math. Anal. Appl., 337 (2008), 36–48. https://doi.org/10.1016/j.jmaa.2007.03.081 doi: 10.1016/j.jmaa.2007.03.081
[26]	A. E. Bashirov, E. Mısırlı, Y. Tandoğdu, A. Özyapıcı, On modeling with multiplicative differential equations, Appl. Math. J. Chin. Univ., 26 (2011), 425–438. https://doi.org/10.1007/s11766-011-2767-6 doi: 10.1007/s11766-011-2767-6
[27]	A. Bashirov, M. Riza, On complex multiplicative differentiation, TWMS J. App. Eng. Math., 1 (2011), 75–85.
[28]	M. Riza, A. Özyapici, E. Misirli, Multiplicative finite difference methods, Quart. Appl. Math., 67 (2009), 745–754. https://doi.org/10.1090/S0033-569X-09-01158-2 doi: 10.1090/S0033-569X-09-01158-2
[29]	K. R. Karthikeyan, G. Murugusundaramoorthy, Properties of a class of analytic functions influenced by multiplicative calculus, Fractal Fract., 8 (2024), 131. https://doi.org/10.3390/fractalfract8030131 doi: 10.3390/fractalfract8030131
[30]	D. Breaz, K. R. Karthikeyan, G. Murugusundaramoorthy, Applications of Mittag–Leffler functions on a subclass of meromorphic functions influenced by the definition of a non-Newtonian derivative, Fractal Fract., 8 (2024), 509. https://doi.org/10.3390/fractalfract8090509 doi: 10.3390/fractalfract8090509
[31]	Q.-H. Xu, Y.-C. Gui, H. M. Srivastava, Coefficient estimates for a certain subclass of analytic and bi-univalent functions, Appl. Math. Lett., 25 (2012), 990–994. https://doi.org/10.1016/j.aml.2011.11.013 doi: 10.1016/j.aml.2011.11.013
[32]	Q.-H. Xu, H.-G. Xiao, H. M. Srivastava, A certain general subclass of analytic and bi-univalent functions and associated coefficient estimate problems, Appl. Math. Comput., 218 (2012), 11461–11465. https://doi.org/10.1016/j.amc.2012.05.034 doi: 10.1016/j.amc.2012.05.034
[33]	P. Zaprawa, On the Fekete-Szego problem for classes of bi-univalent functions, Bull. Belg. Math. Soc. Simon Stevin, 21 (2014), 169–178. https://doi.org/10.36045/bbms/1394544302 doi: 10.36045/bbms/1394544302
[34]	P. Zaprawa, Estimates of initial coefficients for bi-univalent functions, Abstr. Appl. Anal., 2014 (2014), 357480. https://doi.org/10.1155/2014/357480 doi: 10.1155/2014/357480
[35]	K. Sakaguchi, On a certain univalent mapping, J. Math. Soc. Japan, 11 (1959), 72–75. https://doi.org/10.2969/jmsj/01110072 doi: 10.2969/jmsj/01110072
[36]	G. Murugusundaramoorthy, T. Bulboacă, Initial coefficients and Fekete-Szegő inequalities for functions related to van der Pol numbers (VPN), Math. Slovaca, 73 (2023), 1183–1196. https://doi.org/10.1515/ms-2023-0087 doi: 10.1515/ms-2023-0087
[37]	J. Sokół, G. Murugusundaramoorthy, K. Vijaya, On $\lambda$ -pseudo starlike functions associated with vertical strip domain, Asian-Eur. J. Math., 16 (2023), 2350135. https://doi.org/10.1142/S1793557123501358 doi: 10.1142/S1793557123501358

AIMS Mathematics

Bi-univalent functions subordinated to a three leaf function induced by multiplicative calculus

Related Papers:

Abstract

1. Introduction

2. Mathematical construction of the problem

3. Solutions of steady-state concentration profile using the Homotopy Perturbation Method

4. Approximate analytical solutions for the skiin friction, Nusselt and Sherwood numbers

5. Results and discussion

6. Conclusions

Conflict of interest

Acknowledgement

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog