Search Results (54)

Here in this paper, we establish the basic inclusion relations among the harmonic class $\mathbb{HF}(\sigma ,\eta )$ with the classes $S_{\mathcal{HF}}^{*}$ of starlike harmonic functions and $K_{\mathcal{HF}}$ of convex harmonic functions defined in open symmetric unit disk U. Moreover, we investigate inclusion connections for the harmonic classes $\mathcal{T}{\mathfrak{N}}_{\mathbb{HF}}\left(\varrho \right)$ and $\mathcal{T}{\mathfrak{Q}}_{\mathbb{HF}}\left(\varrho \right)$ of harmonic functions by applying the operator $\mathrm{\Lambda }$ associated with the Bessel function. Furthermore, several special cases of the main results are obtained for the particular case $\sigma =0$. Full article

In this paper, the authors derived some new sufficient conditions for q-close-to-convexity with respect to certain functions involving three different normalizations of q-Bessel–Struve functions. These new inequalities, under which the three normalizations of q-Bessel–Struve functions are q-close-to-convex associated with certain functions, hold for $v\ge {\displaystyle \frac{-3}{2}}$ and for all $q\in \left(0,1\right)$. The work is new and has great importance because it shows the pivotal role between the q-special functions and geometric function theory. Full article

In this work, we introduce a new subclass of bi-univalent functions using the $(p,q)$-derivative operator and the concept of subordination to generalized Laguerre polynomials ${\mathbb{L}}_t^{\varsigma }\left(k\right)$, which satisfy the differential equation $k{y}^{″}+(1+\varsigma -k)y^{\prime }+ty=0,$ with $1+\varsigma >0$, $k\in \mathbb{R}$, and $t\ge 0$. We focus on functions that blend the geometric features of starlike and convex mappings in a symmetric setting. The main goal is to estimate the initial coefficients of functions in this new class. Specifically, we obtain sharp upper bounds for $|a_2|$ and $|a_3|$ and for the Fekete–Szegö functional $|a_3-\eta a_2^2|$ for some real number $\eta$. In the final section, we explore several special cases that arise from our general results. These results contribute to the ongoing development of bi-univalent function theory in the context of $(p,q)$-calculus. Full article

This paper presents a novel symmetric q-analogue differential operator designed for meromorphic multivalent functions analytic in the punctured open unit disk. Employing this operator, a new family of meromorphic multivalent functions is proposed and examined in this work. A detailed investigation of this newly defined class of meromorphic multivalent functions is presented, highlighting key geometric characteristics, including sufficiency criteria, coefficient inequalities, distortion and growth behavior, as well as the radii of starlikeness and convexity. Full article

In the present paper, using the q-difference operator, we introduce two classes of q-starlike functions and q-convex functions subordinate to secant hyperbolic functions. As functions in these classes have unique characteristic of missing coefficients on the second term in their analytic expansions, we define a new functional to unify the Hankel determinants with entries of the original coefficients, inverse coefficients, logarithmic coefficients, and inverse logarithmic coefficients for these functions. We obtain the sharp bounds on the new functional for functions in the two classes, and as a consequence, the best results on Hankel determinant for the starlike and convex functions subordinate to secant hyperbolic functions are given. The outcomes include some existing findings as corollaries and may help to deepen the understanding the properties of q-analogue analytic functions. Full article

In this work, we describe the $\mathfrak{q}$-analogue of a multiplier–Ruscheweyh operator of a specific family of linear operators ${\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\tau )$, and we obtain findings related to geometric function theory (GFT) by utilizing approaches established through subordination and knowledge of $\mathfrak{q}$-calculus operators. By using this operator, we develop generalized classes of quasi-convex and close-to-convex functions in this paper. Additionally, the classes ${\mathfrak{K}}_{\mathfrak{q},\rho }^s(\nu ,\tau )\left(\phi \right)$, ${\mathfrak{Q}}_{\mathfrak{q},\rho }^s(\nu ,\tau )\left(\phi \right)$ are introduced. The invariance of these recently formed classes under the $\mathfrak{q}$-Bernardi integral operator is investigated, along with a number of intriguing inclusion relationships between them. Additionally, several unique situations and the beneficial outcomes of these studies are taken into account. Full article

In the present paper, we propose new generalized classes of (p,q)-starlike and (p,q)-convex functions. These classes are introduced by making use of a (p,q)-derivative operator. There are established Fekete–Szegö estimates $|a_3-\mu a_2^2|$ for functions belonging to the newly introduced subclasses. Certain subclasses of analytic univalent functions associated with quasi-subordination are defined. Full article

In this article, we first consider the fractional q-differential operator and the $\left(\lambda ,q\right)$-fractional differintegral operator $D_q^{\lambda }:\mathcal{A}\to \mathcal{A}$. Using the $\left(\lambda ,q\right)$-fractional differintegral operator, we define two new subclasses of analytic functions: the subclass ${\mathcal{S}}^{*}\left(q,\beta ,\lambda \right)$ of starlike functions of order $\beta$ and the class $C_{\mathrm{\Sigma }}^{\lambda ,q}\left(\alpha \right)$ of bi-close-to-convex functions of order $\beta$. We explore the results on coefficient inequality and Fekete–Szegö problems for functions belonging to the class ${\mathcal{S}}^{*}\left(q,\beta ,\lambda \right)$. Using the Faber polynomial technique, we derive upper bounds for the nth coefficient of functions in the class of bi-close-to-convex functions of order $\beta$. We also investigate the erratic behavior of the initial coefficients in the class $C_{\mathrm{\Sigma }}^{\lambda ,q}\left(\alpha \right)$ of bi-close-to-convex functions. Furthermore, we address some known problems to demonstrate the connection between our new work and existing research. Full article

In this work, we study some new applications of symmetric quantum calculus in the field of Geometric Function Theory. We use the cardioid domain and the symmetric quantum difference operator to generate new classes of multivalent q-starlike and q-convex functions. We examine a wide range of interesting properties for functions that can be classified into these newly defined classes, such as estimates for the bounds for the first two coefficients, Fekete–Szego-type functional and coefficient inequalities. All the results found in this research are sharp. A number of well-known corollaries are additionally taken into consideration to show how the findings of this research relate to those of earlier studies. Full article

In mathematical analysis, the q-analogue of a function refers to a modified version of the function that is derived from q-series expansions. This paper is focused on the q-analogue of the exponential function and investigates a class of convex functions associated with it. The main objective is to derive precise inequalities that bound the coefficients of these convex functions. In this research, the initial coefficient bounds, Fekete–Szegő problem, second and third Hankel determinant have been determined. These coefficient bounds provide valuable information about the behavior and properties of the functions within the considered class. Full article

Geometric function theory, a subfield of complex analysis that examines the geometrical characteristics of analytic functions, has seen a sharp increase in research in recent years. In particular, by employing subordination notions, the contributions of different subclasses of analytic functions associated with innovative image domains are of significant interest and are extensively investigated. Since $\Re (1+sinh(z\left)\right)\ngtr 0,$ it implies that the class $S_{sinh}^{*}$ introduced in reference third by Kumar et al. is not a subclass of starlike functions. Now, we have introduced a parameter $\lambda$ with the restriction $0\le \lambda \le ln(1+\sqrt{2}),$ and by doing that, $\Re (1+sinh(\lambda z\left)\right)>0.$ The present research intends to provide a novel subclass of starlike functions in the open unit disk $\mathcal{U},$ denoted as $S_{sinh\lambda }^{*}$, and investigate its geometric nature. For this newly defined subclass, we obtain sharp upper bounds of the coefficients $a_n$ for $n=2,3,4,5.$ Then, we prove a lemma, in which the largest disk contained in the image domain of $q_0\left(z\right)=1+sinh\left(\lambda z\right)$ and the smallest disk containing $q_0\left(\mathcal{U}\right)$ are investigated. This lemma has a central role in proving our radius problems. We discuss radius problems of various known classes, including $S^{*}\left(\beta \right)$ and $\mathcal{K}\left(\beta \right)$ of starlike functions of order $\beta$ and convex functions of order $\beta$. Investigating $S_{sinh\lambda }^{*}$ radii for several geometrically known classes and some classes of functions defined as ratios of functions are also part of the present research. The methodology used for finding $S_{sinh\lambda }^{*}$ radii of different subclasses is the calculation of that value of the radius $r<1$ for which the image domain of any function belonging to a specified class is contained in the largest disk of this lemma. A new representation of functions in this class, but for a more restricted range of $\lambda$, is also obtained. Full article

In this paper, we use the concept of quantum (or $\mathfrak{q}$-) calculus and define a $\mathfrak{q}$-analogous of a fractional differential operator and discuss some of its applications. We consider this operator to define new subclasses of uniformly $\mathfrak{q}$-starlike and $\mathfrak{q}$-convex functions associated with a new generalized conic domain, ${\mathsf{\Lambda }}_{\beta ,\mathfrak{q},\gamma }$. To begin establishing our key conclusions, we explore several novel lemmas. Furthermore, we employ these lemmas to explore some important features of these two classes, for example, inclusion relations, coefficient bounds, Fekete–Szego problem, and subordination results. We also highlight many known and brand-new specific corollaries of our findings. Full article

Our goal in this article is to use ideas from symmetric q-calculus operator theory in the study of meromorphic functions on the punctured unit disc and to propose a novel symmetric q-difference operator for these functions. A few additional classes of meromorphic functions are then defined in light of this new symmetric q-difference operator. We prove many useful conclusions regarding these newly constructed classes of meromorphic functions, such as convolution, subordination features, integral representations, and necessary conditions. The technique presented in this article may be used to produce a wide variety of new types of generalized symmetric q-difference operators, which can subsequently be used to investigate a wide variety of new classes of analytic and meromorphic functions related to symmetric quantum calculus. Full article

In this study, we applied the ideas of subordination and the symmetric q-difference operator and then defined the novel class of bi-univalent functions of complex order $\gamma$. We used the Faber polynomial expansion method to determine the upper bounds for the functions belonging to the newly defined class of complex order $\gamma$. For the functions in the newly specified class, we further obtained coefficient bounds $\left|{\rho }_2\right|$ and the Fekete–Szegő problem $\left|{\rho }_3-{\rho }_2^2\right|$, both of which have been restricted by gap series. We demonstrate many applications of the symmetric Sălăgean q-differential operator using the Faber polynomial expansion technique. The findings in this paper generalize those from previous studies. Full article

This article defines a new class of meromorphic parabolic starlike functions in the punctured unit disc $D^{*}=\{z\in \mathbb{C}:0<|z|<1\}$ that includes fixed second coefficients of class $\mathcal{A}{}_{s,c}^d\left(\psi ,\tau ,\nu ,\eta \right)$ and the q- hypergeometric functions. For the function belonging to the class $\mathcal{A}{}_{s,c}^d\left(\psi ,\tau ,\nu ,\eta \right),$ some properties are obtained, including the coefficient inequalities, closure theorems, and the radius of convexity. Full article