Search

In the recent past, various new subclasses of normalized harmonic functions have been defined in open unit disk U which satisfy second-order and third-order differential inequalities. Here, in this study, we define a new class of normalized harmonic functions in open unit disk U which is satisfying a fourth-order differential inequality. We investigate some useful results such as close-to-convexity, coefficient bounds, growth estimates, sufficient coefficient condition, and convolution for the functions belonging to this new class of harmonic functions. In addition, under convex combination and convolution of its members, we prove that this new class is closed, and we also give some lemmas to prove our main results.

In this paper, we will discuss some generalized classes of analytic functions related with conic domains in the context of $ q $-calculus. In this work, we define and explore Janowski type $ q $-starlike functions in $ q $ -conic domains. We investigate some important properties such as necessary and sufficient conditions, coefficient estimates, convolution results, linear combination, weighted mean, arithmetic mean, radii of starlikeness, growth and distortion results for these classes. It is important to mention that our results are generalization of number of existing results.

In this paper, for the first time, we apply symmetric $ q $ -calculus operator theory to define symmetric Salagean $ q $-differential operator. We introduce a new class $ \widetilde{\mathcal{H}}_{q}^{m}\left(\alpha \right) $ of harmonic univalent functions $ f $ associated with newly defined symmetric Salagean $ q $-differential operator for complex harmonic functions. A sufficient coefficient condition for the functions $ f $ to be sense preserving and univalent and in the same class is obtained. It is proved that this coefficient condition is necessary for the functions in its subclass $ \overline{\widetilde{\mathcal{H}}_{q}^{m}\left(\alpha \right) } $ and obtain sharp coefficient bounds, distortion theorems and covering results. Furthermore, we also highlight some known consequence of our main results.

In this article, we study the Fekete-Szegö functional associated with a new class of analytic functions related to the class of bounded turning by using the principle of quasi-subordination. We derived the coefficient estimates including the classical Fekete-Szegö inequality for functions belonging to this class. We also improved some existing results.

Most subclasses of univalent functions are characterized with functions that map open unit disc ∇ onto the right-half plane. This concept was later modified in the literature with those mappings that conformally map ∇ onto a circular domain. Many researchers were inspired with this modification, and as such, several articles were written in this direction. On this note, we further modify this idea by relating certain subclasses of univalent functions with those that map ∇ onto a sector in the circular domain. As a result, conditions for univalence, radius results, growth rate, and several inclusion relations are obtained for these novel classes. Overall, many consequences of findings show the validity of our investigation.

The main objective of this paper is to study a new family of analytic functions that are q-starlike with respect to m-symmetrical points and subordinate to the q-Janowski function. We investigate inclusion results, sufficient conditions, coefficients estimates, bounds for Fekete–Szego functional |a3−μa22| and convolution properties for the functions belonging to this new class. Several consequences of main results are also obtained.

The investigation of univalent functions is one of the fundamental ideas of Geometric function theory (GFT). However, the class of these functions cannot be investigated as a whole for some particular kind of problems. As a result, the study of its subclasses has been receiving numerous attentions. In this direction, subfamilies of the class of univalent functions that map the open unit disc onto the domain bounded by limacon of Pascal were recently introduced in the literature. Due to the several applications of this domain in Mathematics, Statistics (hypothesis testing problem) and Engineering (rotary fluid processing machines such as pumps, compressors, motors and engines.), continuous investigation of these classes are of interest in this article. To this end, the family of functions for which $ \frac{\varsigma f^{\prime}(\varsigma)}{f(\varsigma)} $ and $ \frac{(\varsigma f^{\prime}(\varsigma))^{\prime}}{f^{\prime}(\varsigma)} $ map open unit disc onto region bounded by limacon are studied. Coefficients bounds, Fekete Szeg $ \ddot{ \rm{o}} $ inequalities and the bounds of the third Hankel determinants are derived. Furthermore, the sharp radius for which the classes are linked to each other and to the notable subclasses of univalent functions are found. Finally, the idea of subordination is utilized to obtain some results for functions belonging to these classes.

In this paper, we investigate the geometric properties of Jackson and Hahn-Exton $ q $-Bessel functions and perform their normalization for the analyticity in open unit disk $ E $. By applying normalized Jackson and Hahn-Exton $ q $-Bessel functions and idea of convolution we introduce a new operator and define new family of subclasses of analytic functions related with generalized conic domain. For these subclasses of analytic functions, we investigate inclusion relations and integral preserving properties. Also we will use $ q $-Bernardi integral operator to discuss some applications of our main results.

Recently, a stabilizer free weak Galerkin (SFWG) method is proposed in [ 14 ], which is easier to implement and more efficient. In this paper, we developed an SFWG scheme for solving the general second-order elliptic problem on triangular meshes in 2D. This new SFWG method will dramatically reduce the error between the $ L^{2} $-projection of the exact solution and the numerical solution.

In this paper our aim is to study some valuable problems dealing with newly defined subclass of multivalent $ q $-starlike functions. These problems include the initial coefficient estimates, Toeplitz matrices, Hankel determinant, Fekete-Szego problem, upper bounds of the functional $ \left \vert a_{p+1}-\mu a_{p+1}^{2}\right \vert $ for the subclass of multivalent $ q $-starlike functions. As applications we study a $ q $-Bernardi integral operator for a subclass of multivalent $ q $-starlike functions. Furthermore, we also highlight some known consequence of our main results.

In this paper, we are mainly interested to study the generalization of typically real functions in the unit disk. We study some coefficient inequalities concerning this class of functions. In particular, we find the Zalcman conjecture for generalized typically real functions.

In this paper, the concepts of quantum (or q-) calculus and conic regions are combined to define a new domain Ωk, q, γ which represents the generalized conic regions. Then by using a certain generalized conic domain Ωk, q, γ we define and investigate a new subclass of normalized analytic functions in open unit disk E. We also investigate a number of useful properties and characteristics of this subclass such as, structural formula, necessary and sufficient condition, coefficient estimates, Feketo-Szego problem, distortion inequalities, closure theorem, and subordination result. We also highlight some known consequences of our main results as corollaries.

In the present paper, we investigate a majorization problem for the class $% M_{\alpha, \beta }^{\nu, j}(\eta, \varkappa; A, B)$ of meromorphic functions and the class $N_{\alpha, \beta }^{\nu, j}(\theta, b;A, B)$ of meromorphic spirllike functions related with a convolution operator. We extend the results existing in literature for higher order derivative. Several consequences of the main results in the form of corollaries are also pointed out.

In this paper, we define a new subclass $k$-$\mathcal{Q} (\alpha)$ of analytic functions, which generalizes the class of $k$-uniformly convex functions. Various interesting relationships between $k$-$\mathcal{Q} (\alpha)$ and the class $\mathcal{B}(\delta)$ of functions with bounded turning are derived.

The quantum (or q -) calculus is widely applied in various operators which include the q -difference ( q -derivative) operator, and this operator plays an important role in geometric function theory (GFT) as well as in the theory of hypergeometric series. In our present investigation, we introduce and study q -differential operator associated with q -Mittag–Leffler function which is an extension of the Salagean q -differential operator. By using this newly defined operator, we define a new subclass of analytic function and studied certain subclass of analytic function in generalized conic domain Ω k , q , γ . For this class, we investigate structural formula, coefficient estimates, sufficient condition, Fekete–Szegö problem, and also some subordination results.

Geometric function theory combines geometric tools and their applications for information and communication analysis. It is also successfully used in the field of advanced signals, image processing, machine learning, speech and sound recognition. Various new subclasses of analytic functions have been defined using quantum calculus to investigate many interesting properties of these subclasses. This article defines a new class of q-starlike functions in the open symmetric unit disc ∇ using symmetric quantum calculus. Extreme points for this class as well as coefficient estimates and closure theorems have been investigated. By fixing several coefficients finitely, all results were generalized into families of analytic functions.

Making use of Ruscheweyh q-differential operator, we define a new subclass of uniformly convex functions and corresponding subclass of starlike functions with negative coefficients. The main object of this paper is to obtain, coefficient estimates, closure theorems and extreme point for the functions belonging to this new class. The results are generalized to families with fixed finitely many coefficients.

The second order elliptic equation, which is also know as the diffusion-convection equation, is of great interest in many branches of physics and industry. In this paper, we use the weak Galerkin finite element method to study the general second order elliptic equation. A weak Galerkin finite element method is proposed and analyzed. This scheme features piecewise polynomials of degree $k\geq 1$ on each element and piecewise polynomials of degree $k-1\geq 0$ on each edge or face of the element. Error estimates of optimal order of convergence rate are established in both discrete $H^1$ and standard $L^2$ norm. The paper also presents some numerical experiments to verify the efficiency of the method.

In our present investigation, we first introduce several new subclasses of analytic and bi-univalent functions by using a certain $q$-integral operator in the open unit disk $$\mathbb{U}=\{z: z\in \mathbb{C} \quad \text{and} \quad \left \vert z\right \vert

In this paper, the concepts of symmetric q-calculus and conic regions are used to define a new domain Ωk,q,α˜, which generalizes the symmetric conic domains. By using the domain Ωk,q,α˜, we define a new subclass of analytic and q-starlike functions in the open unit disk U and establish some new results for functions of this class. We also investigate a number of useful properties and characteristics of this subclass, such as coefficients estimates, structural formulas, distortion inequalities, necessary and sufficient conditions, closure and subordination results. The proposed approach is also compared with some existing methods to show the reliability and effectiveness of the proposed methods.

In the present investigation, subclasses of analytic functions with respect to symmetrical points which are defined by the generalized Bessel functions of the first kind of orderμare introduced. Furthermore, some alluring geometric properties of these classes, which include inclusion property, integral-preserving properties, coefficients, and distortion results are studied. Moreover, some consequences of our results are also given.

The aim of present investigation is to study a new class of analytic function related with the Sokol-Nunokawa class. We derived relationships of this class with strongly starlike functions and obtained many interesting results.

In this article, by using the concept of the quantum (or q-) calculus and a general conic domain &Omega, k , q , we study a new subclass of normalized analytic functions with respect to symmetrical points in an open unit disk. We solve the Fekete-Szeg&ouml, type problems for this newly-defined subclass of analytic functions. We also discuss some applications of the main results by using a q-Bernardi integral operator.

Recently, there is a rapid increase of research in the area of Quantum calculus (known as q -calculus) due to its widespread applications in many areas of study, such as geometric functions theory. To this end, using the concept of q -conic domains of Janowski type as well as q - calculus, new subclasses of analytic functions are introduced. This family of functions extends the notion of α -convex and quasi-convex functions. Furthermore, a coefficient inequality, sufficiency criteria, and covering results for these novel classes are derived. Besides, some remarkable consequences of our investigation are highlighted.

In the present paper, we introduce and investigate the Fekete-Szego functional associated with a new subclass of ¨ analytic functions, which we have defined here by using the principle of quasi-subordination between analytic functions. Some sufficient conditions for functions belonging to this class are also derived. The results presented here improve and generalize several known results.

In our present investigation, by using Salagean q -differentialoperator we introduce and define new subclass$k-\mathcal{US}(q,\gamma ,m),$ $\gamma \in C\backslash \{0\},$ andstudied certain subclass of analytic functions in conic domains. Weinvestigate the number of useful properties of this class suchstructural formula and coefficient estimates Fekete--Szego problem,we give some subordination results, and some other corollaries.

In our present investigation, we use the technique of convolution and quantum calculus to study the Salagean q-differential operator. By using this operator and the concept of the Janowski function, we define certain new classes of analytic functions. Some properties of these classes are discussed, and numerous sharp results such as coefficient estimates, distortion theorem, radii of star-likeness, convexity, close-to-convexity, extreme points, and integral mean inequalities of functions belonging to these classes are obtained and studied.

In recent years, applications of Bessel differential equations have been commonly used in univalent functions theory. The main object of the present paper is to give some characteristic properties for some subclasses of uniformly starlike and convex functions which are defined here by means of the normalized form of the generalized Bessel function to be univalent in the open unit disc. Furthermore, we also establish some results of these subclasses related to a particular integral operator. Some corresponding consequences of our main results are also considered.

Certain new classes of q-convex and q-close to convex functions that involve the q-Janowski type functions have been defined by using the concepts of quantum (or q-) calculus as well as q-conic domain Ω k , q [ λ , α ] . This study explores some important geometric properties such as coefficient estimates, sufficiency criteria and convolution properties of these classes. A distinction of new findings with those obtained in earlier investigations is also provided, where appropriate.

Let &Sigma, be the class of meromorphic functions f of the form f ( &zeta, ) = &zeta, + &sum, n = 0 &infin, a n &zeta, &minus, n which are analytic in &Delta, = { &zeta, &isin, C : | &zeta, &gt, 1 } . For n &isin, N 0 : = N &cup, { 0 } , the nth Faber polynomial &Phi, n ( w ) of f &isin, &Sigma, is a monic polynomial of degree n that is generated by a function &zeta, f &prime, ( &zeta, ) / ( f ( &zeta, ) &minus, w ) . For given f &isin, by F n , i ( f ) , we denote the ith coefficient of &Phi, n ( w ) . For given 0 &le, &alpha, &lt, 1 and 0 &lt, &beta, &le, 1 , let us consider domains H &alpha, and S &beta, &sub, C defined by H &alpha, = { w &isin, C : Re ( w ) &gt, } and S &beta, C : | arg ( w ) | &lt, } , which are symmetric with respect to the real axis. A function f &isin, is called meromorphic starlike of order &alpha, if &zeta, ) / f ( &zeta, ) &isin, H &alpha, for all &zeta, &Delta, Another function f &isin, is called meromorphic strongly starlike of order &beta, S &beta, In this paper we investigate the sharp bounds of F n , n &minus, i ( f ) , n &isin, N 0 , i &isin, { 2 , 3 , 4 } , for meromorphic starlike functions of order &alpha, and meromorphic strongly starlike of order &beta, Similar estimates for meromorphic convex functions of order &alpha, ( 0 &le, 1 ) and meromorphic strongly convex of order &beta, ( 0 &lt, 1 ) are also discussed.

By making use of q-calculus, we define and investigate several new subclasses of bi-univalent mappings related to the q-Noor integral operator. The coefficient bounds | u 2 | , | u 3 | and the Fekete&ndash, Szegő problem u 3 &minus, &mu, u 2 2 for mappings belonging to these classes are derived.

The aim of the present paper is to introduce a new class of analytic functions by using a q-integral operator in the conic region. It is worth mentioning that these regions are symmetric along the real axis. We find the coefficient estimates, the Fekete&ndash, Szeg&ouml, inequality, the sufficiency criteria, the distortion result, and the Hankel determinant problem for functions in this class. Furthermore, we study the inverse coefficient estimates for functions in this class.

Recently, Hussain [Hussain, Saqib (2010). Some applications of Miller-Mocanu lemma on certain classes of meromorphic functions, \textquotedblright Appl. Math. Comput.\textquotedblright , 216(2010), 3016-3021], introduced certain new classes of meromorphic functions using newly defined convolution operators. The purpose of present investigation is to extend the work of Hussain by considering some new classes of meromorphic functions. We derive some inclusion relationships, coefficient bound, integral representation and extreme point theorem.