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The results of this work belong to the field of geometric function theory, being based on differential subordination methods. Using the idea of the q-calculus operators, we define the q-analogue of the multiplier- Ruscheweyh operator of a specific family of linear operators, Isq,µ(λ,l). Our major goal is to build and investigate some analytic function subclasses using Isq,µ(λ, l). Also, some differential subordination and superordination results are obtained. Moreover, based on the new theoretical results, several examples are constructed. For every differential superordination under investigation, the best subordinant is provided. [ABSTRACT FROM AUTHOR]

In this investigation, we study a class of analytic functions of type Carathéodory style in the open unit disk connected with some fractal domains. This class of analytic functions is formulated based on a kind of Langevin differential equations (LDEs). We aim to study the analytic solvability of LDEs in the advantage of geometric function theory consuming the geometric properties of the Julia fractal (JF) and other fractal connected with the logarithmic function. The analytic solutions of the LDEs are obtainable by employing the subordination theory. [ABSTRACT FROM AUTHOR]

Orthogonal polynomials have been widely employed by renowned authors within the context of geometric function theory. This study is driven by prior research and aims to address the —Fekete-Szegö problem. Additionally, we provide bound estimates for the coefficients and an upper bound estimate for the second Hankel determinant for functions belonging to the category of analytical and bi-univalent functions. This investigation incorporates the utilization of Euler polynomials. [ABSTRACT FROM AUTHOR]

This work presents a novel investigation that utilizes the integral operator I p , λ n in the field of geometric function theory, with a specific focus on sandwich theorems. We obtained findings about the differential subordination and superordination of a novel formula for a generalized integral operator. Additionally, certain sandwich theorems were discovered. [ABSTRACT FROM AUTHOR]

The results of this work have a connection with the geometric function theory and they were obtained using methods based on subordination along with information on q -calculus operators. We defined the q -analogue of multiplier- Ruscheweyh operator of a certain family of linear operators I q , μ s (λ , ℓ) f (ς) (s ∈ N 0 = N ∪ { 0 } , N = { 1 , 2 , 3 ,.. } ; ℓ , λ , μ ≥ 0 , 0 < q < 1) . Our major goal was to build some analytic function subclasses using I q , μ s (λ , ℓ) f (ς) and to look into various inclusion relationships that have integral preservation features. The results of this work have a connection with the geometric function theory and they were obtained using methods based on subordination along with information on -calculus operators. We defined the -analogue of multiplier- Ruscheweyh operator of a certain family of linear operators . Our major goal was to build some analytic function subclasses using and to look into various inclusion relationships that have integral preservation features. [ABSTRACT FROM AUTHOR]

This study presents a subclass S (β) of bi-univalent functions within the open unit disk region D . The objective of this class is to determine the bounds of the Hankel determinant of order 3, ( Ⱨ 3 (1) ). In this study, new constraints for the estimates of the third Hankel determinant for the class S (β) are presented, which are of considerable interest in various fields of mathematics, including complex analysis and geometric function theory. Here, we define these bi-univalent functions as S (β) and impose constraints on the coefficients │ a n │ . Our investigation provides the upper bounds for the bi-univalent functions in this newly developed subclass, specifically for n = 2, 3, 4, and 5. We then derive the third Hankel determinant for this particular class, which reveals several intriguing scenarios. These findings contribute to the broader understanding of bi-univalent functions and their potential applications in diverse mathematical contexts. Notably, the results obtained may serve as a foundation for future investigations into the properties and applications of bi-univalent functions and their subclasses. [ABSTRACT FROM AUTHOR]

The field of geometric function theory has thoroughly investigated starlike functions concerning symmetric points. The main objective of this work is to derive certain geometric properties, such as the starlikeness of order δ , convexity of order δ , k-starlikeness, k-uniform convexity, lemniscate starlikeness and convexity, exponential starlikeness and convexity, and pre-starlikeness for the Galué type Struve function (GTSF). Furthermore, the conditions for GTSF belonging to the Hardy space are also derived. The results obtained in this work generalize several results available in the literature. [ABSTRACT FROM AUTHOR]

One of the fundamental parts of Geometric Function Theory is the study of analytic functions in different domains with critical geometrical interpretations. This article defines a new generalized domain obtained based on the quotient of two analytic functions. We derive various properties of the new class of normalized analytic functions X defined in the new domain, including the sharp estimates for the coefficients a 2 , a 3 , and a 4 , and for three second-order and third-order Hankel determinants, H 2 , 1 X , H 2 , 2 X , and H 3 , 1 X . The optimality of each obtained estimate is given as well. [ABSTRACT FROM AUTHOR]

Over the last few years, Geometric Function Theory (GFT) as one of the most prime branch of complex analysis has gained a considerable and an impressive attention from many researchers, largely because it deals with the study of the geometric properties of analytic functions and their numerous applications in various fields of mathematics such as in special functions, probability distributions, and fractional calculus. The investigations in this paper are on two new classes of analytic functions defined in the unit disk ε = {z ∈ C : |z| < 1} and denoted by XSq (b,K) and XTq (b,K). Function f in the classes satisfy the conditions f(0) = f ′(0) -- 1 = 0, hence can be of series type f(z) = z + a2z² + a3z³ + …, z ∈ ε. The definition of the two new classes of analytic functions embed some well-known special functions such as the Galuê-type Struve function, modified error function and a starlike function whose coefficients are Bell's numbers while some involving mathematical principles are the q-derivative, inequalities, convolution and subordination. The main results from these classes are however, the upper estimates of some initial bounds such as |an| (n = 2, 3, 4) and the Fekete-Szegö functional |a3 - øa2/2| (ø ∈ C) of functions f ∈ XSq (b,K) and f ∈ XTq (b,K). [ABSTRACT FROM AUTHOR]

The results from this paper are related to the geometric function theory. In order to obtain them, we use the technique based on the properties of the differential subordination and superordination one of the newest techniques used in this field, we obtain some differential subordination and superordination results for multivalent functions defined by differintegral operator with j-derivatives =p(v, p;l) f (z) for l > 0; v; p 2 R; such that (p - j) ≥ 0; v > lp; (p 2 N) in the open unit disk U. Differential sandwich result is also obtained. Also, the results are followed by some special cases and counter examples. [ABSTRACT FROM AUTHOR]

Gegenbauer polynomials constitute a numerical tool that has attracted the interest of many function theorists in recent times mainly due to their real-life applications in many areas of the sciences and engineering. Their applications in geometric function theory (GFT) have also been considered by many researchers. In this paper, this powerful tool is associated with the prolific concepts of convolution and subordination. The main purpose of the research contained in this paper is to introduce and study a new subclass of analytic functions. This subclass is presented using an operator defined as the convolution of the generalized distribution and the error function and applying the principle of subordination. Investigations into this subclass are considered in connection to Carathéodory functions, the modified sigmoid function and Bell numbers to obtain coefficient estimates for the contained functions. [ABSTRACT FROM AUTHOR]

Some well-known authors have extensively used orthogonal polynomials in the framework of geometric function theory. We are motivated by the previous research that has been conducted and, in this study, we solve the Fekete–Szegö problem as well as give bound estimates for the coefficients and an upper bound estimate for the second Hankel determinant for functions in the class  G Σ (v , σ)  of analytical and bi-univalent functions, implicating the Euler polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we discuss and introduce a new study using an integral operator w k , μ m in geometric function theory, especially sandwich theorems. We obtained some conclusions for differential subordination and superordination for a new formula generalized integral operator. In addition, certain sandwich theorems were found. The differential subordination theory's features and outcomes are symmetric to those derived using the differential subordination theory. [ABSTRACT FROM AUTHOR]

The results contained in this paper are the result of a study regarding fractional calculus combined with the classical theory of differential subordination established for analytic complex valued functions. A new operator is introduced by applying the Libera integral operator and fractional integral of order λ for analytic functions. Many subordination properties are obtained for this newly defined operator by using famous lemmas proved by important scientists concerned with geometric function theory, such as Eenigenburg, Hallenbeck, Miller, Mocanu, Nunokawa, Reade, Ruscheweyh and Suffridge. Results regarding strong starlikeness and convexity of order α are also discussed, and an example shows how the outcome of the research can be applied. [ABSTRACT FROM AUTHOR]

The results from this paper are related to the geometric function theory. In order to obtain them, we use the technique based on differential subordination, one of the newest techniques used in the field, also known as the technique of admissible functions. For that, the appropriate classes of admissible functions are first defined. Based on these classes, we obtain some differential subordination and superordination results for multivalent meromorphic functions, analytic in the punctured unit disc, related to a linear operator ℑ ρ , τ p (ν) , for τ > 0 , ν , ρ ∈ C , such that R e (ρ − ν) ≧ 0 , R e (ν) > τ p , (p ∈ N) . Moreover, taking into account both subordination and superordination results, we derive a sandwich-type theorem. The connection with some other known results and an example are also provided. [ABSTRACT FROM AUTHOR]

We define a new class of exponential starlike functions constructed by a linear operator involving normalized form of the generalized Struve function. Making use of a technique of differential subordination introduced by Miller and Mocanu, we investigate several new results related to the Briot–Bouquet differential subordinations for the linear operator involving the normalized form of the generalized Struve function. We also obtain univalent solutions to the Briot–Bouquet differential equations and observe that these solutions are the best dominant of the Briot–Bouquet differential subordinations for the exponential starlike function class. Moreover, we give an application of fractional integral operator for a complex-valued function associated with the generalized Struve function. The significance of this paper is due to the technique employed in proving the results and novelty of these results for the Struve functions. The approach used in this paper can lead to several new problems in geometric function theory associated with special functions. [ABSTRACT FROM AUTHOR]

The topics studied in the geometric function theory of one variable functions are connected with the concept of Symmetry because for some special cases the analytic functions map the open unit disk onto a symmetric domain. Thus, if all the coefficients of the Taylor expansion at the origin are real numbers, then the image of the open unit disk is a symmetric domain with respect to the real axis. In this paper, we formulate the q-differential operator associated with the q-Raina function using quantum calculus, that is the so-called Jacksons' calculus. We establish a new subclass of analytic functions in the unit disk by using this newly developed operator. The theory of differential subordination inspired our approach; therefore, we geometrically explore the most popular properties of this new operator: subordination properties, coefficient bounds, and the Fekete-Szegő problem. As special cases, we highlight certain well-known corollaries of our primary findings. [ABSTRACT FROM AUTHOR]

Ever since Lotfi A. Zadeh published the paper "Fuzzy Sets" in 1965 setting the basis of a new theory named fuzzy sets theory, many scientists have developed this theory and its applications. Mathematicians were especially interested in extending classical mathematical results in the fuzzy context. Such an extension was also done relating fuzzy sets theory and geometric theory of analytic functions. The study begun in 2011 has many interesting published outcomes and the present paper follows the line of the previous research in the field. The aim of the paper is to give some references related to the connections already made between fuzzy sets theory and geometric theory of analytic functions and to present some new results that might prove interesting for mathematicians willing to enlarge their views on certain aspects of the merge between the two theories. Using the notions of fuzzy differential subordination and the classical notion of differential subordination for analytic functions, two criteria for the univalence of the analytic functions are stated in this work. [ABSTRACT FROM AUTHOR]

Fractional differential operators are mostly investigated for functions of real variables. In this paper, we present two fractional differential operators for a class of normalized analytic functions in the open unit disk. The suggested operators are investigated according to concepts in geometric function theory, using the concepts of convexity and starlikeness. Therefore, we reformulate the new operators in the Ma-Minda class of analytic functions, in order to act on normalized analytic functions. Our method is based on subordination, superordination, and majorization theory. As an application, we employ these operators to generalize Bernoulli’s equation and a special class of Briot-Bouquet equations. The solution of the generalized equation is formulated by a hypergeometric function. [ABSTRACT FROM AUTHOR]

Recently, q-analogue of Noor integral operator and other special operators became importance in the field of Geometric Function Theory. In this study, by connecting this operators we introduced an interesting class of bi-univalent functions and obtained coefficient estimates for this new class. Also we present some relevant corollaries. [ABSTRACT FROM AUTHOR]

The central purpose of this effort is to investigate analytic and geometric properties of a class of normalized analytic functions in the open unit disk involving Bernoulli's formula. As a consequence, some solutions are indicated by the well-known hypergeometric function. The class of starlike functions is investigated containing the suggested class. [ABSTRACT FROM AUTHOR]

In this note, we formulate a new linear operator given by Airy functions of the first type in a complex domain. We aim to study the operator in view of geometric function theory based on the subordination and superordination concepts. The new operator is suggested to define a class of normalized functions (the class of univalent functions) calling the Airy difference formula. As a result, the suggested difference formula joining the linear operator is modified to different classes of analytic functions in the open unit disk. [ABSTRACT FROM AUTHOR]

In this present paper, we obtain some differential subordination and superordination results, by using generalized operators for certain subclass of analytic functions in the open unit disk. Also, we derive some sandwich results. [ABSTRACT FROM AUTHOR]

Recently, there has been a vivid interest on special functions from the point of view of geometric function theory. Geometric properties of special functions like univalency, starlikeness and convexity appear in works of many mathematicians. In this paper, firstly, we obtain a new family of integral operators involving normalized Mittag-Leffler functions. Then, we give various sufficient conditions for convexity of this integral operator in the open unit disk. Several consequences of the main results are also shown. [ABSTRACT FROM AUTHOR]

In this work, we introduce and investigate an interesting operator Q based on fractional derivative which is introduced by Owa and Srivastava in [10]. We consider a new technique to prove our results and then, we introduce two subclasses of analytic functions in the open unit disk U concerning with this operator. Some results such as inclusion relations, subordination properties, integral preserving properties and argument estimate are investigated. [ABSTRACT FROM AUTHOR]

For analytic functions f (z) = z + Σ∞k=2fkzk and g(z) = z + Σ∞k=2gkz k=2 k=2 in the unit disk properties of the Hadamard compositions Df [Sf * Dn [s]g and Dn [r] f * Dnl [R]g of their Gelfond-Leont'ev-Salagean derivatives Dnl,]f(z) =z+Σ∞k=2 (k) fkzk n and Gelfond-Leont'ev-Ruscheweyh derivatives Dln[R]f(z) = z + Σ∞k=2 fkZk ln+k -1 are investigated. For study, generalized orders are used. A connection between the growth of the maximal term of the Hadamard composition of Gelfond-Leont'ev-Saalaagean derivatives or Gelfond-Leont'ev-Ruscheweyh derivatives and the growth of the maximal term of these derivatives of Hadamard composition is established. Similar results are obtained in terms of the classical order and the lower order of the growth. [ABSTRACT FROM AUTHOR]

Quantum calculus (the calculus without limit) appeared for the first time in fluid mechanics, noncommutative geometry and combinatorics studies. Recently, it has been included into the field of geometric function theory to extend differential operators, integral operators, and classes of analytic functions, especially the classes that are generated by convolution product (Hadamard product). In this effort, we aim to introduce a quantum symmetric conformable differential operator (Q-SCDO). This operator generalized some well-know differential operators such as Sàlàgean differential operator. By employing the Q-SCDO, we present subclasses of analytic functions to study some of its geometric solutions of q-Painlevé differential equation (type III). [ABSTRACT FROM AUTHOR]

For analytic functions f(z) normalized by f(0) = f(0)-1 = 0 in the open unit disk U, a class Ak(β1, β2, β3, β4, λ) of function f(z) defined by(β1z/(Dkf(z)z)+ 2z2(Dkf(z)"+β3z3(Dkf(z)/z= β4z4(Dkf(z)/z""≤λ(zϵU),where Dk denote the Ruscheweyh di erential operator of order k, k ϵ N0 := N U{0} for somecomplex number numbers β1,β2, β3, β4 and for some real λ > 0 is introduced. The object of thepresent paper is to obtain some interesting properties of function class Ak( β1< β2, β3, β4, λ). Our results generalize the results due to Uyaníik et al. (2010) and Uyaníkík and Owa (2011). [ABSTRACT FROM AUTHOR]

The object of the present investigation is to solve Fekete-Szegö problem for a new class Vλp (a, c, A,B) of p-valent analytic functions involving the Saitoh operator in the unit disk. We also obtain subordination results and some interesting corollaries for functions in Ap involving this operator. Relevant connections of the results obtained here with those given by earlier workers on the subject are also mentioned. [ABSTRACT FROM AUTHOR]

We define V ( α, β) ( α < 1 and β > 1), the new subclass of analytic functions with bounded positive real part, $$V\left( {\alpha ,\beta } \right): = \left\{ {f \in A:\alpha < \operatorname{Re} \left\{ {{{\left( {\frac{z}{{f\left( z \right)}}} \right)}^2}f'\left( z \right)} \right\} < \beta } \right\}$$ , and study some properties of V ( α, β). We also study the class U ( γ) ( γ > 0): $$u\left( \gamma \right): = \left\{ {f \in A:\left| {{{\left( {\frac{z}{{f\left( z \right)}}} \right)}^2}f'\left( z \right)} \right| - 1 < \gamma } \right\}$$ , where A is the class of normalized functions. [ABSTRACT FROM AUTHOR]

Let k - ST[A, B], k ≥ 0, -1 ≤ B < A ≤ 1 be the class of normalized analytic functions defined in the open unit disk satisfying R((B-1)zf'(z)/f(z)-(A-1)/(B+1)zf'(z)/f(z)-(A+1)) > k/(B-1)zf'(z)/f(z)-(A-1)/(B+1)zf'(z)/f(z)-(A+1)-1/ and let k - UCV [A,B], k ≥ 0, -1 ≤ B < A ≤ 1 be the corresponding class satisfying R((B-1)zf'(z)/f(z)-(A-1)/(B+1)zf'(z)/f(z)-(A+1)) > k/(B-1)zf'(z)/f(z)-(A-1)/(B+1)zf'(z)/f(z)-(A+1)-1/. For an appropriate δ > 0, the δ neighborhood of a function f ∊ k -UCV[A, B] is shown to consist of functions in the class k - ST[A, B]. [ABSTRACT FROM AUTHOR]

In the present paper we introduce a new subclass of analytic functions. We prove a sharp upper bound to the second Hankel determinant associated with the kth root transform [ f(zk) ]1/k of the normalized analytic function f(z), when it belongs to this class, using Toeplitz determinants. [ABSTRACT FROM AUTHOR]

Let f : (ℝn, 0) → (ℝ, 0) be a nonconstant analytic function defined in a neighborhood of the origin 0 ∈ ℝn. The classical Łojasiewicz inequality states that there exist positive constants δ, c and l such that |f(x)| ≥ cd(x, f-1(0))l for |x| ≤ δ, where d(x, f-1(0)) denotes the distance from x to the set f-1(0). The Łojasiewicz exponent of f at the origin 0 ∈ ℝn, denoted by , is the infimum of the exponents l satisfying the Łojasiewicz inequality. In this paper, we establish a formula for computing the Łojasiewicz exponent of f in terms of the Newton polyhedron of f in the case where f is nonnegative and nondegenerate. [ABSTRACT FROM AUTHOR]