Applications of a certain $q$-integral operator to the subclasses of analytic and bi-univalent functions

Khan, Bilal; Srivástava, H. M.; Tahir, Muhammad; Darus, Maslina; Ahmad, Qazi Zahoor; Khan, Nazar

doi:10.3934/math.2021061

Cited by 57 publications

(27 citation statements)

References 44 publications

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“…The special families examined in this research paper using Al-Oboudi type operator could inspire further research related to other aspects such as families using q-derivative operator [22], [35], meromorphic bi-univalent function families associated with Al-Oboudi differential operator [30] and families using integro-differential operators [27].…”

Section: Discussionmentioning

confidence: 94%

Coefficient Bounds for Al-Oboudi Type Bi-univalent Functions based on a Modified Sigmoid Activation Function and Horadam Polynomials

Swamy¹

2021

EJMS

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Using the Al-Oboudi type operator, we present and investigate two special families of bi-univalent functions in $\mathfrak{D}$, an open unit disc, based on $\phi(s)=\frac{2}{1+e^{-s} },\,s\geq0$, a modified sigmoid activation function (MSAF) and Horadam polynomials. We estimate the initial coefficients bounds for functions of the type $g_{\phi}(z)=z+\sum\limits_{j=2}^{\infty}\phi(s)d_jz^j$ in these families. Continuing the study on the initial cosfficients of these families, we obtain the functional of Fekete-Szeg\"o for each of the two families. Furthermore, we present few interesting observations of the results investigated.

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Section: Discussionmentioning

confidence: 94%

Coefficient Bounds for Al-Oboudi Type Bi-univalent Functions based on a Modified Sigmoid Activation Function and Horadam Polynomials

Swamy¹

2021

EJMS

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“…Moreover, the subclasses of q-starlike functions associated with the Janwoski or some other functions have been studied by the many authors (see, for example, [13][14][15][16][17][18][19][20]). For some more recent investigations based upon the q-calculus, we may refer the interested reader to the works in [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38]. Our present research is a continuation of some of these earlier developments.…”

Section: Introduction Definitions and Motivationmentioning

confidence: 86%

A Subclass of Multivalent Janowski Type q-Starlike Functions and Its Consequences

Srivástava

Ahmad³

et al. 2021

Symmetry

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In this article, by utilizing the theory of quantum (or q-) calculus, we define a new subclass of analytic and multivalent (or p-valent) functions class Ap, where class Ap is invariant (or symmetric) under rotations. The well-known class of Janowski functions are used with the help of the principle of subordination between analytic functions in order to define this subclass of analytic and p-valent functions. This function class generalizes various other subclasses of analytic functions, not only in classical Geometric Function Theory setting, but also some q-analogue of analytic multivalent function classes. We study and investigate some interesting properties such as sufficiency criteria, coefficient bounds, distortion problem, growth theorem, radii of starlikeness and convexity for this newly-defined class. Other properties such as those involving convex combination are also discussed for these functions. In the concluding part of the article, we have finally given the well-demonstrated fact that the results presented in this article can be obtained for the (p,q)-variations, by making some straightforward simplification and will be an inconsequential exercise simply because the additional parameter p is obviously unnecessary.

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“…Recently, in his survey-cum-expository review article, Srivastava [46] demonstrated how the theories of the basic (or q-) calculus and the fractional q-calculus have significantly encouraged and motivated further developments in Geometric Function Theory of Complex Analysis, which also include such topics of analytic and bi-univalent functions as those involving (for example) q-starlike, q-convex, and other related q-function classes as well as the associated Fekete-Szegö type inequalities (see, for details, [38,39,47,48,49,50]).…”

Section: Concluding Remarks and Observationsmentioning

confidence: 99%

Faber polynomial coefficient estimates of bi-close-to-convex functions connected with the Borel distribution of the Mittag-Leffler type

2021

J. Nonlinear Var. Anal.

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By using the Borel distribution series of the Mittag-Leffler type, we introduce a new class of the bi-close-to-convex functions defined in the open unit disk. We then apply the Faber polynomial expansion method in order to investigate the estimates for the general Taylor-Maclaurin coefficients of the functions belonging to this new class of bi-close-to-convex functions in the open unit disk. We consider the Fekete-Szegö type inequalities for the bi-close-to-convex function class and also consider several corollaries and the consequences of the results presented in this paper. Keywords. Analytic, univalent and bi-univalent functions; Starlike, convex, close-to-convex and biclose-to-convex functions; Mittag-Leffler type Borel distribution; Faber polynomial expansion method; Fekete-Szegö type inequalities.

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Applications of a certain $q$-integral operator to the subclasses of analytic and bi-univalent functions

Cited by 57 publications

References 44 publications

Coefficient Bounds for Al-Oboudi Type Bi-univalent Functions based on a Modified Sigmoid Activation Function and Horadam Polynomials

Coefficient Bounds for Al-Oboudi Type Bi-univalent Functions based on a Modified Sigmoid Activation Function and Horadam Polynomials

A Subclass of Multivalent Janowski Type q-Starlike Functions and Its Consequences

Faber polynomial coefficient estimates of bi-close-to-convex functions connected with the Borel distribution of the Mittag-Leffler type

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