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Definitions and Concepts
Published in Daniel Zwillinger, Vladimir Dobrushkin, Handbook of Differential Equations, 2021
Daniel Zwillinger, Vladimir Dobrushkin
The q-derivative is defined by Dqf(x)=ddxqf(x)=f(qx)−f(x)(q−1)x=f(x+ϵx)−f(x)ϵx
Some Subclasses of Analytic Functions and Their Properties
Published in Michael Ruzhansky, Hemen Dutta, Advanced Topics in Mathematical Analysis, 2019
Nizami Mustafa, Veysel Nezir, Hemen Dutta
Also, the coefficient problems, representation formula and distortion theorems for these subclasses S*(α, β, μ) and C*(α, β, μ) of the analytic functions were given by Owa and Aouf in Owa and Aouf (1988). In Kadioğlu (2003), results of Silverman were extended by Kadioğlu. Researches on subclasses of analytic functions have been of interest for many mathematicians. In this field, recently different type of subclasses have been introduced and properties of analytic functions with some conditions have been studied. The editor of this book, Hemen Dutta is also involved in many works in the field of special functions and like his recent joint studies (“S. O. Olatunji and H. Dutta. Fekete-Szegö problem for certain analytic functions defined by q-derivative operator with respect to other points, submitted” and “S. O. Olatunji, H. Dutta, and M. A. Adeniyi. Subclasses of bi-Sakaguchi function associated with q-difference operator. Thai J. Math., in editing.”), there is a crucial interest in the field for the others.
Accelerated gradient methods with absolute and relative noise in the gradient
Published in Optimization Methods and Software, 2023
Artem Vasin, Alexander Gasnikov, Pavel Dvurechensky, Vladimir Spokoiny
We start by recalling a way to calculate a derivative in a general Hilbert space. Let , where is the unique solution of the equation . Assume that the partial q-derivative of the operator is invertible. Then, we have Therefore, The same result can be obtained by considering the Lagrange functional with Indeed, by simple calculations, we can connect these two approaches by setting Next, we demonstrate this technique on an inverse problem based on an elliptic initial-boundary-value problem. Let be the solution of the following problem, which we refer to as (P) Here we use subscripts x, y to denote the corresponding partial derivatives. The first two relations constitute the system of equations , and the last two ones constitute the feasible set Q.
Reciprocity of degenerate poly-Dedekind-type DC sums
Published in Applied Mathematics in Science and Engineering, 2023
Lingling Luo, Yuankui Ma, Wenpeng Zhang, Taekyun Kim
Jackson [11] first defined the q-derivative operator . In recent years, the q-derivatives have been widely used in other polynomials, such as, Khan et al. [12] defined a number of subclasses of q-starlike functions, which associated with Janowski functions, and some coefficient inequalities with them. Shi et al. [13] defined a new subclass of Janowski-type multivalent q-starlike functions. Zhang et al. [14] defined a new subclass of bi-univalent functions and obtained some interesting results. By using q-Poisson distribution, Khan et al. [15] defined function classes and derived some new subclasses and some useful results.
Minimizing the difference of two quasiconvex functions over a vector-valued quasiconvex system
Published in Optimization, 2020
Stephan Dempe, Nazih Abderrazzak Gadhi, Khadija Hamdaoui
Using the Q-subdifferential introduced by Suzuki and Kuroiwa [2], together with a special constraint qualification, we give necessary optimality conditions to Some reasons of choosing the Q-derivative are that it generalizes the subdifferential in convex analysis and the Gâteau-derivative for smooth functions, it also has some chain rules such as the Q-subdifferential of the composite and the Q-subdifferential of the supremum of quasi-convex functions which can be interesting in the study of this type of problems [3]. The obtained results are very similar to those of Hiriart-Urruty [4] for DC (difference of convex) problems.