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.

In this study, we introduce a new class of normalized analytic and bi-univalent functions denoted by ${\mathcal{D}}_{\Sigma }(\delta ,\eta ,\lambda ,t,r)$. These functions are connected to the Bazilevič functions and the $\lambda$-pseudo-starlike functions. We employ Sakaguchi Type Functions and Horadam polynomials in our survey. We establish the Fekete-Szegö inequality for the functions in ${\mathcal{D}}_{\Sigma }(\delta ,\eta ,\lambda ,t,r)$ and derive upper bounds for the initial Taylor–Maclaurin coefficients $|a_2|$ and $|a_3|$. Additionally, we establish connections between our results and previous research papers on this topic.

Many researchers have defined the q-analogous of differential and integral operators for analytic functions using the concept of quantum calculus in the geometric function theory. In this study, we conduct a comprehensive investigation to identify the uses of the Sălăgean q-differential operator for meromorphic multivalent functions. Many features of functions that belong to geometrically defined classes have been extensively studied using differential operators based on q-calculus operator theory. In this research, we extended the idea of the q-analogous of the Sălăgean differential operator for meromorphic multivalent functions using the fundamental ideas of q-calculus. With the help of this operator, we extend the family of Janowski functions by adding two new subclasses of meromorphic q-starlike and meromorphic multivalent q-starlike functions. We discover significant findings for these new classes, including the radius of starlikeness, partial sums, distortion theorems, and coefficient estimates.

Let κ be an analytic function in the open unit disc $\mathcal{U}$ in the complex plane with $\mathrm{\kappa }(0)=1$. Then $\mathrm{\kappa }\in \mathcal{K}(\mathrm{\lambda })$ if and only if $\mathrm{\kappa }(t)\prec {\varphi }_{\mathrm{\lambda }}(t)$, where ≺ is a subordination relation and the function ${\varphi }_{\mathrm{\lambda }}(t):={(1+t)}^{\mathrm{\lambda }}$, for $0<\mathrm{\lambda }<1$, maps $\mathcal{U}$ to a symmetric domain about the real axis in the right-half plane. This article finds the sharp radii of starlikeness for three classes of normalized analytic functions in the open unit disc. It also finds the solutions to first-order differential equations and utilizes these results together with the differential subordination developed by Miller and Mocanu to obtain the conditions so that some first-order differential relations associated with ${\varphi }_{\mathrm{\lambda }}$ are subordinate to certain Ma and Minda functions. Consequently, it provides sufficient conditions for the normalized analytic functions to belong to some established subclasses of starlike functions.

This paper proves several new inequalities for the Euclidean operator radius, which refine some recent results. It is shown that the new results are much more accurate than the related, recently published results. Moreover, inequalities for both symmetric and non-symmetric Hilbert space operators are studied.

In the literature, there are many criteria to generalize the concept of a numerical radius; one of the most recent and interesting generalizations is the so-called generalized Euclidean operator radius, which reads: ${\omega }_p\left(T_1,\cdots ,T_n\right):=\underset{∥x∥=1}{sup}{\left({\displaystyle \sum _{i=1}^n}{\left|⟨T_{i}x,x⟩\right|}^p\right)}^{1/p},p\ge 1,$ for all Hilbert space operators $T_1,\cdots ,T_n$. Simply put, it is the numerical radius of multivariable operators. This study establishes a number of new inequalities, extensions, and generalizations for this type of numerical radius. More precisely, by utilizing the mixed Schwarz inequality and the extension of Furuta’s inequality, some new refinement inequalities are obtained for the numerical radius of multivariable Hilbert space operators. In the case of $n=1$, the resulting inequalities could be considered extensions and generalizations of the classical numerical radius.