Operators, inequalities and approximation : theory and applications

The book collects chapters on operator theory, as well as related approximation results and analytic inequalities. It discusses the properties of various types of operators, methods for approximating such operators, proximity point problems, applications of approximation methods in other fields such as engineering, and some analytic inequalities. It seeks to capture both the pure and applied aspects of the topics discussed. Several of the concepts covered in the book are fundamental to many aspects of applied science and engineering. The intriguing and novel aspect of the book is that it focuses on foundational aspects of the topics as well as reasonable application ideas and inputs useful information for practical applications in a variety of other scientific and engineering fields. Dedication Preface Contents 1 Approximation by a Double Sequence of Operators Involving Multivariable qq-Lagrange–Hermite Polynomials 1.1 Introduction 1.2 Bivariate Lagrange–Hermite Operators 1.3 Rate of Convergence 1.4 Generalization of German upper R Subscript n 1 comma q Sub Subscript n 1 Subscript comma alpha Sub Superscript left parenthesis 1 right parenthesis Subscript comma ellipsis comma alpha Sub Superscript left parenthesis r 1 right parenthesis Subscript Superscript n 2 comma q Super Subscript n 2 Superscript comma beta Super Superscript left parenthesis 1 right parenthesis Superscript comma ellipsis comma beta Super Superscript left parenthesis r 2 right parenthesismathfrakRn1,qn1,α(1),...,α(r1)n2,qn2,β(1),...,β(r2) 1.5 GBS (Generalized Boolean Sum) Operators 1.6 Future Scope of Work 1.6.1 Kantorovich Type Operators Involving Multivariable Q-Lagrange–Hermite Polynomials 1.6.2 Kantorovich–Taylor Type Operators Involving Multivariable Q-Lagrange–Hermite Polynomials 1.6.3 GBS-Kantorovich Type Operators Involving Multivariable Q-Lagrange–Hermite Polynomials References 2 Approximation Process of the Fuzzy Meyer-König and Zeller Operators 2.1 Introduction 2.2 Some Auxilary Results 2.3 Some Fuzzy Korovkin-Type Approximation Results 2.4 Consequence References 3 On Approximation of Signals in the Generalized Zygmund Class Using left parenthesis upper E comma s right parenthesis left parenthesis ModifyingAbove upper N With quotation dash comma q Subscript n Baseline right parenthesis(E,s) (overlineN,qn) Mean 3.1 Introduction 3.2 Known Results 3.3 Main Theorems 3.4 Lemmas 3.5 Proof of the Lemmas 3.6 Proof of Main Theorems 3.7 Conclusion References 4 Trigonometric Approximation of Signals Belonging to upper L i p left parenthesis xi left parenthesis t right parenthesis comma r right parenthesisLip(ξ(t), r) Class by left parenthesis upper C comma 1 right parenthesis left parenthesis upper N comma p Subscript m Baseline comma q Subscript m Baseline right parenthesis left parenthesis upper E comma theta right parenthesis(C, 1)(N,pm,qm)(E,θ) Means of Conjugate Fourier Series 4.1 Introduction 4.2 Known Results 4.2.1 Theorem 4.2.2 Theorem 4.2.3 Theorem 4.3 Useful Lemmas 4.3.1 Lemma 4.3.2 Lemma 4.3.3 Lemma ch426 4.3.4 Lemma ch426 4.4 Main Results 4.4.1 Theorem 4.4.2 Theorem 4.4.3 Theorem 4.5 Corollaries 4.5.1 Corollary 4.5.2 Corollary 4.5.3 Corollary 4.6 Conclusion 4.7 Applications and Uses References 5 Turán Type Inequalities For the left parenthesis p comma k right parenthesis minus(p,k)-Generalization of the Mittag-Leffler Function 5.1 Introduction 5.2 Pochhammer Symbol and Gamma Function with Two-Parameter 5.2.1 Two-Parameter Pochhammer Symbol 5.2.2 Two-Parameter Gamma Function 5.3 Generalizations of the Mittag-Leffler Function 5.3.1 The Mittag-Leffler Function 5.3.2 The left parenthesis p comma k right parenthesis minus(p,k)-Mittag-Leffler Function 5.4 Turán Type Inequalities 5.4.1 Turán Type Inequalities for the Mittag-Leffler Function 5.5 Main Results 5.6 Concluding Remarks References 6 Multiplicative Generalized Hardy-Rogers-Type bold italic upper FF-Proximal Non-self Mappings and Best Proximity Point Approximation 6.1 Introduction 6.2 Preliminaries 6.3 Best Proximity Point Theorems for Multiplicative Generalized bold italic upper F Subscript script upper P upper CFmathcalPC 6.4 Best Proximity Point Approximation for Multiplicative Khammahawong and Kumam's bold italic upper FF-Contraction Mapping References 7 Best Proximity Point Problems in G-Metric Spaces and Its Applications 7.1 Introduction 7.2 G-Metric Spaces 7.3 Best Proximity Points in G-Metric Spaces 7.4 Application to Fixed Point Results 7.5 Application to Solution of Functional Equations 7.6 Conclusion References 8 On a New Subclass of Bi-Univalent Analytic Functions Characterized by left parenthesis script upper P comma script upper Q right parenthesis(mathcalP,mathcalQ)-Lucas Polynomial Coefficients via Sălăgean Differential Operator 8.1 Main Concepts and Properties of Univalent Functions 8.1.1 Conformal Mappings 8.1.2 The Classes of Univalent and Bi-Univalent Functions 8.1.3 Coefficient Estimates 8.1.4 Convex and Starlike Functions 8.1.5 Functions with Positive Real Part 8.2 Coefficient Estimates for the Class upper Q Subscript normal upper Sigma Baseline left parenthesis m comma n semicolon normal upper Theta right parenthesisQΣ(m,n;Θ) 8.2.1 Basic Concepts 8.2.2 The Class upper Q Subscript normal upper Sigma Baseline left parenthesis m comma n semicolon normal upper Theta right parenthesisQΣ(m,n;Θ) 8.2.3 Fekete–Szegö Inequalities References 9 Sufficient Conditions for Generalized Integral Operators Involving the Rabotnov Function 9.1 Introduction 9.2 A Set of Lemmas 9.3 Univalence and Convexity Conditions for the Integral Operator in (9.5) 9.4 Univalence and Convexity Conditions for the Integral Operator in (9.6) 9.5 Conclusions References Discusses topics in operator approximation, proximity point problems and fractal generation Includes interesting and novel topics of analytic inequalities Focusses on the properties of various types of operators and their applications