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In this present paper, we define a new operator in conjugation with the basic (or q-) calculus. We then make use of this newly defined operator and define a new class of analytic and bi-univalent functions associated with the q-derivative operator. Furthermore, we find the initial Taylor–Maclaurin coefficients for these newly defined function classes of analytic and bi-univalent functions. We also show that these bounds are sharp. The sharp second Hankel determinant is also given for this newly defined function class. [ABSTRACT FROM AUTHOR]

In recent years, many new subclasses of analytic and bi-univalent functions have been studied and examined from different viewpoints and prospectives. In this article, we introduce new subclass of analytic and bi-univalent functions based on Mittag-Leffler type Borel distribution associated with the Gegenbauer polynomials. Furthermore we obtain estimates for |a2|, |a3|, and |a4| coefficients and Fekete-Szegö inequality for this functions class. Providing specific values to parameters involved in our main results, we get some new results. [ABSTRACT FROM AUTHOR]

In recent years, using the idea of analytic and bi-univalent functions, many ideas have been developed by different well-known authors, but the using Gegenbauer polynomials along with certain bi-univalent functions is very rare in the literature. We are essentially motivated by this recent research going on, here in our present investigation, we make use of certain q-derivative operator and Gegenbauer polynomials and define a new subclass of analytic and bi-univalent functions. We then obtain certain coefficient bounds, the Fekete–Szegö inequalities and upper bounds for the second-order Hankel determinant for the defined functions class. [ABSTRACT FROM AUTHOR]

In recent years, many authors have studied and investigated majorization results for different subclasses of analytic functions. In this paper, we give some majorization results for certain non vanishing analytic functions, whose ratios are subordinated to different domains in the open unit disk. [ABSTRACT FROM AUTHOR]

In mathematics, physics, and engineering, orthogonal polynomials and special functions play a vital role in the development of numerical and analytical approaches. This field of study has received a lot of attention in recent decades, and it is gaining traction in current fields, including computational fluid dynamics, computational probability, data assimilation, statistics, numerical analysis, and image and signal processing. In this paper, using q-Hermite polynomials, we define a new subclass of bi-univalent functions. We then obtain a number of important results such as bonds for the initial coefficients of | a 2 | , | a 3 | , and | a 4 | , results related to Fekete–Szegö functional, and the upper bounds of the second Hankel determinant for our defined functions class. [ABSTRACT FROM AUTHOR]

In this paper, using the basic concepts of symmetric q -calculus operator theory, we define a symmetric q -difference operator for m -fold symmetric functions. By considering this operator, we define a new subclass ℛ b φ , m , q of m -fold symmetric bi-univalent functions in open unit disk U. As in applications of Faber polynomial expansions for f m ∈ ℛ b φ , m , q , we find general coefficient a m k + 1 for n ≥ 4 , Fekete–Szegő problems, and initial coefficients a m + 1 and a 2 m + 1 . Also, we construct q -Bernardi integral operator for m -fold symmetric functions, and with the help of this newly defined operator, we discuss some applications of our main results. For validity of our result, we have chosen to give some known special cases of our main results in the form of corollaries and remarks. [ABSTRACT FROM AUTHOR]

By making use of some families of integral and derivative operators, many distinct subclasses of analytic, starlike functions, and symmetric starlike functions have already been defined and investigated from numerous perspectives. In this article, with the help of the one-parameter Bernardi integral operator, we investigate several majorization results for the class of normalized starlike functions, which are associated with the Janowski functions. We also give some particular cases of our main results. Finally, we direct the interested readers to the possibility of examining the fundamental or quantum (or q -) extensions of the findings provided in this work in the concluding section. However, the (p , q) -variations of the suggested q -results will provide relatively minor and inconsequential developments because the additional (rather forced-in) parameter p is obviously redundant. [ABSTRACT FROM AUTHOR]

By using the basic (or q)-Calculus many subclasses of analytic and univalent functions have been generalized and studied from different viewpoints and perspectives. In this paper, we aim to define certain new subclasses of an analytic function. We then give necessary and sufficient conditions for each of the defined function classes. We also study necessary and sufficient conditions for a function whose coefficients are probabilities of q-Poisson distribution. To validate our results, some known consequences are also given in the form of Remarks and Corollaries. [ABSTRACT FROM AUTHOR]

In recent years, the Hankel determinant bounds for different subclasses of analytic, starlike and symmetric starlike functions have been discussed and studied by the many well-known authors. In this paper, we first consider a new subclass of analytic function and then we derive the fourth Hankel determinant bound for this class. [ABSTRACT FROM AUTHOR]

In our present investigation, by applying q -calculus operator theory, we define some new subclasses of m -fold symmetric analytic and bi-univalent functions in the open unit disk U = z ∈ ℂ : z < 1 and use the Faber polynomial expansion to find upper bounds of a m k + 1 and initial coefficient bounds for a m + 1 and a 2 m + 1 as well as Fekete-Szego inequalities for the functions belonging to newly defined subclasses. Also, we highlight some new and known corollaries of our main results. [ABSTRACT FROM AUTHOR]

A new class of q-close-to-convex functions associated with Janowski functions is defined. In this regard, we give sufficient conditions and prove famous de Branges theorem for this newly-defined class of q-close-to-convex functions. We also give the application of our results to finding sufficient conditions for the celebrated Mittag-Leffler function to be a Janowski q-close-to-convex function. [ABSTRACT FROM AUTHOR]

In this paper we introduce and investigate two new subclasses HΣq (α, λ) and HΣq(β, λ) of analytic and bi-univalent functions in the open unit disk U . Furthermore, we find non-sharp estimates on the first two Taylor-Maclaurin coefficients |a2| and |a3| for functions in these new subclasses. [ABSTRACT FROM AUTHOR]

In this paper, we introduce certain new subclass of bi-univalent functions in an open unit disk associated with generalized Hypergeometric function. By using Faber polynomial expansions to find a general coefficient bounds ∣an∣ for n ⩾ 3 of class of bi-subordinate functions subject to a gap series condition, also find initial coefficients bounds. [ABSTRACT FROM AUTHOR]

This paper deals with a new subclass of univalent function associated with the right half of the lemniscate of Bernoulli. We find the upper bound of the Hankel determinant H 3 (1) for this subclass by applying the Carlson–Shaffer operator to it. The present work also deals with certain properties of this newly defined subclass, such as the upper bound of the Hankel determinant of order 3, coefficient estimates, etc. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a normalized analytic (symmetric under rotation) function, f, in an open unit disk that satisfies the condition ℜ f z g z > 0 , for some analytic function, g, with ℜ z + 1 − 2 n z g z > 0 , ∀ n ∈ N. We calculate the radius constants for different classes of analytic functions, including, for example, for the class of star-like functions connected with the exponential functions, i.e., the lemniscate of Bernoulli, the sine function, cardioid functions, the sine hyperbolic inverse function, the Nephroid function, cosine function and parabolic star-like functions. The results obtained are sharp. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of harmonic univalent functions with respect to k-symmetric points by using a newly-defined q-analog of the derivative operator for complex harmonic functions. For this harmonic univalent function class, we derive a sufficient condition, a representation theorem, and a distortion theorem. We also apply a generalized q-Bernardi–Libera–Livingston integral operator to examine the closure properties and coefficient bounds. Furthermore, we highlight some known consequences of our main results. In the concluding part of the article, we have finally reiterated the well-demonstrated fact that the results presented in this article can easily be rewritten as the so-called (p , q) -variations by making some straightforward simplifications, and it will be an inconsequential exercise, simply because the additional parameter p is obviously unnecessary. [ABSTRACT FROM AUTHOR]

First, by making use of the concept of basic (or q-) calculus, as well as the principle of subordination between analytic functions, generalization R q (h) of the class R (h) of analytic functions, which are associated with the leaf-like domain in the open unit disk U , is given. Then, the coefficient estimates, the Fekete–Szegö problem, and the second-order Hankel determinant H 2 (1) for functions belonging to this class R q (h) are investigated. Furthermore, similar results are examined and presented for the functions z f (z) and f − 1 (z) . For the validity of our results, relevant connections with those in earlier works are also pointed out. [ABSTRACT FROM AUTHOR]

In this paper, the authors introduce a new subclass of meromorphic q-starlike functions which are associated with the Janowski functions. A characterization of meromorphically q-starlike functions associated with the Janowski functions has been obtained when the coefficients in their Laurent series expansion about the origin are all positive. This leads to a study of coefficient estimates, distortion theorems, partial sums, and the radius of starlikeness estimates for this class. It is seen that the class considered demonstrates, in some respects, properties analogous to those possessed by the corresponding class of univalent analytic functions with negative coefficients. [ABSTRACT FROM AUTHOR]

By making use of the concept of basic (or q-) calculus, various families of q-extensions of starlike functions of order α in the open unit disk U were introduced and studied from many different viewpoints and perspectives. In this paper, we first investigate the relationship between various known classes of q-starlike functions that are associated with the Janowski functions. We then introduce and study a new subclass of q-starlike functions that involves the Janowski functions. We also derive several properties of such families of q-starlike functions with negative coefficients including (for example) distortion theorems. [ABSTRACT FROM AUTHOR]

Various operators of fractional calculus, as well as their quantum (or q -) extensions have been used widely and successfully in the study of the Taylor-Maclaurin coefficient estimation problems for many different families of normalized analytic, univalent and bi-univalent functions in Geometric Function Theory of Complex Analysis. On the other hand, Numerous writers have extensively employed orthogonal polynomials in the framework of Geometric Function Theory in Complex analysis. The recent and ongoing studies on this topic is primarily what drives us. In this paper, we solve the Fekete-Szegö problem for the symmetric function classes of analytical and bi-univalent functions involving the Euler polynomials. We also give estimates for the coefficients and an upper bound for the second Hankel determinant. [ABSTRACT FROM AUTHOR]