Generalized Briot-Bouquet Differential Equation Based on New Differential Operator with Complex Connections

Abstract

A class of Briot–Bouquet differential equations is a magnificent part of investigating the geometric behaviors of analytic functions, using the subordination and superordination concepts. In this work, we aim to formulate a new differential operator with complex connections (coefficients) in the open unit disk and generalize a class of Briot–Bouquet differential equations (BBDEs). We study and generalize new classes of analytic functions based on the new differential operator. Consequently, we define a linear operator with applications.

Entire solutions of a class of algebraic Briot–Bouquet differential equations utilizing majority concept

“…ϕ (z) = 1 + a 1 z + (a 1 + a 2 )z 2 + (a 1 + 2a 2 )z 3 + (a 1 + 3a 2 )z 4 + (a 1 + 4a 2 )z 5 + O z 6 The next result indicates the value of constant coefficients of ℵ k ϕ when ϕ is starlike in U.…”

“…The most important property of ABBDE is that it shows that every meromorphic solution of ABBDE belongs to the class of meromorphic functions containing elliptic functions and their degenerates, i.e., the rational functions of one exponential e ζ z , ζ ∈ C, z ∈ U = {z ∈ C : |z| < 1} and rational functions. Many consequences for higher-order ABBDE are also presented in a sequence of documents (see [1][2][3][4][5][6][7][8][9][10][11][12]).…”

“…+ (5a 1 + 20a 2 + 21a 3 + 8a 4 + a 5 )z 5 + O z 6 , ℵ 6 ϕ (z) = 1 + a 1 z + (2a 1 + a 2 )z 2 + (3a 1 + 4a 2 + a 3 )z 3 + (4a 1 + 10a 2 + 6a 3 + a 4 )z 4 + (5a 1 + 20a 2 + 21a 3 + 8a 4 + a 5 )z 5 Abbreviations ABBDE, Algebraic Briot-Bouquet differential equations.…”

On Special Differential Subordinations Using Fractional Integral of Sălăgean and Ruscheweyh Operators

“…Using operators derived as a combination of those two operators, interesting results can be obtained, as it can be observed in [9][10][11][12]. We can also refer to [13,14] for applications of differential operators in the analysis of phenomena from mathematical biology and physics.…”

Geometric properties of mixed operator involving Ruscheweyh derivative and Salagean operator

“…Lately, Lupas [10] presented a amalgamation of two well-known differential operators prearranged by Ruscheweyh [12] and Sȃlȃgean [13]. Later, these operators are investigated by researchers considering different classes and formulas of analytic functions [5,8].…”