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The book contains papers published in the Mathematics Special Issue, entitled "Geometrical Theory of Analytic Functions". Fifteen papers devoted to the study concerning complex-valued functions of one variable present new outcomes related to special classes of univalent functions, differential equations in view of geometric function theory, quantum calculus and its applications in geometric function theory, operators and special functions associated with differential subordination and superordination theories and starlikeness, and convexity criteria.

Research & information: general --- Mathematics & science --- univalent function --- conformable fractional derivative --- subordination and superordination --- analytic function --- open unit disk --- bi-univalent functions --- Hadamard (convolution) product --- coefficients bounds --- q-derivative operator --- differential subordination --- lacunary function --- gap function --- centered polygonal numbers --- natural boundary --- singularities --- broom topology --- convex function --- starlike function --- dominant --- best dominant --- analytic functions --- univalent functions --- Taylor–Maclaurin series --- holomorphic function --- p-valent function --- convolution product --- ξ-Generalized Hurwitz–Lerch Zeta function --- admissible functions --- strongly close-to-convex functions --- starlike functions --- meromorphic strongly starlike functions --- Sălăgean integral and differential operator --- coefficient bounds --- Fekete–Szegő problem --- Janowski functions --- subordination --- cosine hyperbolic function --- q-difference operator --- subordinating factor sequence --- bounded analytic functions of complex order --- q-generalized linear operator --- Painlevé differential equation --- symmetric solution --- asymptotic expansion --- symmetric conjugate points --- horadam polynomial --- Fekete–Szegö problem --- differential superordination --- harmonic function --- subordinant --- best subordinant --- meromorphic functions --- Hurwitz–Lerch Zeta-function --- Riemann zeta function --- differential inclusions --- differential containments --- differential inequalities --- differential subordinations

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This volume consists of a collection of 14 accepted submissions (including several invited feature articles) to the Special Issue of MDPI's journal Symmetry on the general subject area of integral transformations, operational calculus and their applications from many different parts around the world. The main objective of the Special Issue was to gather review, expository, and original research articles dealing with the state-of-the-art advances in integral transformations and operational calculus as well as their multidisciplinary applications, together with some relevance to the aspect of symmetry. Various families of fractional-order integrals and derivatives have been found to be remarkably important and fruitful, mainly due to their demonstrated applications in numerous diverse and widespread areas of mathematical, physical, chemical, engineering, and statistical sciences. Many of these fractional-order operators provide potentially useful tools for solving ordinary and partial differential equations, as well as integral, differintegral, and integro-differential equations; fractional-calculus analogues and extensions of each of these equations; and various other problems involving special functions of mathematical physics and applied mathematics, as well as their extensions and generalizations in one or more variables.

History of engineering & technology --- Stancu-type Bernstein operators --- Bézier bases --- Voronovskaja-type theorems --- modulus of continuity --- rate of convergence --- bivariate operators --- approximation properties --- statistical convergence --- P-convergent --- statistically and relatively modular deferred-weighted summability --- relatively modular deferred-weighted statistical convergence --- Korovkin-type approximation theorem --- modular space --- convex space --- N-quasi convex modular --- N-quasi semi-convex modular --- vehicle collaborative content downloading --- fuzzy comprehensive evaluation --- VANET --- delay differential equations --- integral operator --- periodic solutions --- subordinations --- exponential function --- Hankel determinant --- fractional differential equations with input --- Mittag-Leffler stability --- left generalized fractional derivative --- ρ-Laplace transforms --- functional integral equations --- Banach algebra --- fixed point theorem --- measure of noncompactness --- Geometric Function Theory --- q-integral operator --- q-starlike functions of complex order --- q-convex functions of complex order --- (δ,q)-neighborhood --- meromorphic multivalent starlike functions --- subordination --- univalent function --- symmetric differential operator --- unit disk --- analytic function --- analytic functions --- conic region --- Hadamard product --- differential subordination --- differential superordination --- generalized fractional differintegral operator --- Convex function --- Simpson’s rule --- differentiable function --- weights --- positive integral operators --- convolution operators --- n/a --- Bézier bases --- Simpson's rule

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This volume consists of a collection of 14 accepted submissions (including several invited feature articles) to the Special Issue of MDPI's journal Symmetry on the general subject area of integral transformations, operational calculus and their applications from many different parts around the world. The main objective of the Special Issue was to gather review, expository, and original research articles dealing with the state-of-the-art advances in integral transformations and operational calculus as well as their multidisciplinary applications, together with some relevance to the aspect of symmetry. Various families of fractional-order integrals and derivatives have been found to be remarkably important and fruitful, mainly due to their demonstrated applications in numerous diverse and widespread areas of mathematical, physical, chemical, engineering, and statistical sciences. Many of these fractional-order operators provide potentially useful tools for solving ordinary and partial differential equations, as well as integral, differintegral, and integro-differential equations; fractional-calculus analogues and extensions of each of these equations; and various other problems involving special functions of mathematical physics and applied mathematics, as well as their extensions and generalizations in one or more variables.

Stancu-type Bernstein operators --- Bézier bases --- Voronovskaja-type theorems --- modulus of continuity --- rate of convergence --- bivariate operators --- approximation properties --- statistical convergence --- P-convergent --- statistically and relatively modular deferred-weighted summability --- relatively modular deferred-weighted statistical convergence --- Korovkin-type approximation theorem --- modular space --- convex space --- N-quasi convex modular --- N-quasi semi-convex modular --- vehicle collaborative content downloading --- fuzzy comprehensive evaluation --- VANET --- delay differential equations --- integral operator --- periodic solutions --- subordinations --- exponential function --- Hankel determinant --- fractional differential equations with input --- Mittag-Leffler stability --- left generalized fractional derivative --- ρ-Laplace transforms --- functional integral equations --- Banach algebra --- fixed point theorem --- measure of noncompactness --- Geometric Function Theory --- q-integral operator --- q-starlike functions of complex order --- q-convex functions of complex order --- (δ,q)-neighborhood --- meromorphic multivalent starlike functions --- subordination --- univalent function --- symmetric differential operator --- unit disk --- analytic function --- analytic functions --- conic region --- Hadamard product --- differential subordination --- differential superordination --- generalized fractional differintegral operator --- Convex function --- Simpson’s rule --- differentiable function --- weights --- positive integral operators --- convolution operators --- n/a --- Bézier bases --- Simpson's rule

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In recent years, various families of fractional-order integral and derivative operators, such as those named after Riemann-Liouville, Weyl, Hadamard, Grunwald-Letnikov, Riesz, Erdelyi-Kober, Liouville-Caputo, and so on, have been found to be remarkably important and fruitful, due mainly to their demonstrated applications in numerous seemingly diverse and widespread areas of the mathematical, physical, chemical, engineering, and statistical sciences. Many of these fractional-order operators provide interesting, potentially useful tools for solving ordinary and partial differential equations, as well as integral, differintegral, and integro-differential equations; fractional-calculus analogues and extensions of each of these equations; and various other problems involving special functions of mathematical physics and applied mathematics, as well as their extensions and generalizations in one or more variables. For this Special Issue, we invite and welcome review, expository, and original research articles dealing with the recent advances in the theory of fractional-order integral and derivative operators and their multidisciplinary applications.

Research & information: general --- Mathematics & science --- Legendre wavelet --- operational matrix --- systems of fractional order differential equations --- Liouville_Caputo sense --- analytic functions --- starlike functions --- univalent functions --- length problems --- SIR model --- Hermite collocation method --- approximate solution --- Hermite polynomials and series --- collocation points --- starlike and q-starlike functions --- q-derivative operator --- q-hypergeometric functions --- conic and generalized conic domains --- Hankel determinant --- Toeplitz matrices --- fractional differential equations --- Riemann-stieltjes integral --- monotone iterative method --- upper and lower solutions --- fractional diffusion equation --- controllability --- impulsive system --- unique continuation property --- non-linear fractional variational problems --- orthogonal polynomials --- Rayleigh-Ritz method --- error analysis --- convergence analysis --- Banach space --- Sobolev space --- Laplace operators --- nonlocal boundary conditions --- fractional differential systems --- mixed Riemann–Liouville integral --- mixed Hadamard integral --- Hilfer derivative --- Hadamard derivative --- coupled system --- random solution --- variable order fractional derivative --- initial value problem --- piecewise constant functions --- starlike function --- Toeplitz determinant --- sine function --- upper bound --- convex functions --- subordination --- logarithmic coefficients --- Gauss hypergeometric function --- confluent hypergeometric function --- Appell’s functions --- incomplete fractional calculus --- Riemann-Liouville fractional integral --- generating functions --- fractional integrals --- fractional derivatives --- Mittag–Leffler functions --- field spectrum --- fractional calculus --- desert soil --- available potassium --- correlation analysis --- Post-Exposure Prophylaxis --- latent reservoir --- HIV infection --- fractional order model --- multivalent functions --- close-to-convex functions --- uniformly starlike functions --- uniformly close-to-convex functions --- conic domains --- q-calculus --- differential subordination --- Janowski function --- cardioid domain --- limacon domain --- fractional order unknown input fuzzy observer --- fractional order Takagi–Sugeno models --- L2 optimization --- linear matrix inequalities --- unmeasurable premise variables --- global solutions --- fractional in time and space shallow-water system --- ψ-Caputo fractional derivative --- Hilfer fractional derivative --- Ulam stability --- pantograph differential equation --- nonlocal integral condition --- Miura transform --- soliton equations --- logarithm --- n/a --- mixed Riemann-Liouville integral --- Appell's functions --- Mittag-Leffler functions --- fractional order Takagi-Sugeno models

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Researches and investigations involving the theory and applications of integral transforms and operational calculus are remarkably wide-spread in many diverse areas of the mathematical, physical, chemical, engineering and statistical sciences.

infinite-point boundary conditions --- nonlinear boundary value problems --- q-polynomials --- ?-generalized Hurwitz–Lerch zeta functions --- Hadamard product --- password --- summation formulas --- Hankel determinant --- multi-strip --- Euler numbers and polynomials --- natural transform --- fuzzy volterra integro-differential equations --- zeros --- fuzzy differential equations --- Szász operator --- q)-Bleimann–Butzer–Hahn operators --- distortion theorems --- analytic function --- generating relations --- differential operator --- pseudo-Chebyshev polynomials --- Chebyshev polynomials --- Mellin transform --- uniformly convex functions --- operational methods --- differential equation --- ?-convex function --- Fourier transform --- q)-analogue of tangent zeta function --- q -Hermite–Genocchi polynomials --- Dunkl analogue --- derivative properties --- q)-Euler numbers and polynomials of higher order --- exact solutions --- encryption --- spectrum symmetry --- advanced and deviated arguments --- PBKDF --- wavelet transform of generalized functions --- fuzzy general linear method --- Lommel functions --- highly oscillatory Bessel kernel --- generalized mittag-leffler function --- audio features --- the uniqueness of the solution --- analytic --- Mittag–Leffler functions --- Dziok–Srivastava operator --- Bell numbers --- rate of approximation --- Bessel kernel --- univalent functions --- inclusion relationships --- Liouville–Caputo-type fractional derivative --- tangent polynomials --- Bernoulli spiral --- multi-point --- q -Hermite–Euler polynomials --- analytic functions --- Fredholm integral equation --- orthogonality property --- Struve functions --- cryptography --- Janowski star-like function --- starlike and q-starlike functions --- piecewise Hermite collocation method --- uniformly starlike and convex functions --- q -Hermite–Bernoulli polynomials --- generalized functions --- meromorphic function --- basic hypergeometric functions --- fractional-order differential equations --- q -Sheffer–Appell polynomials --- integral representations --- Srivastava–Tomovski generalization of Mittag–Leffler function --- Caputo fractional derivative --- Bernoulli --- symmetric --- sufficient conditions --- nonlocal --- the existence of a solution --- functions of bounded boundary and bounded radius rotations --- differential inclusion --- symmetry of the zero --- recurrence relation --- nonlinear boundary value problem --- Volterra integral equations --- Ulam stability --- q)-analogue of tangent numbers and polynomials --- starlike function --- function spaces and their duals --- strongly starlike functions --- q)-Bernstein operators --- vibrating string equation --- ?-generalized Hurwitz-Lerch zeta functions --- bound on derivatives --- Janowski convex function --- volterra integral equation --- strongly-starlike function --- Hadamard product (convolution) --- regular solution --- generalized Hukuhara differentiability --- functions with positive real part --- exponential function --- q–Bleimann–Butzer–Hahn operators --- Carlitz-type q-tangent polynomials --- distributions --- Carlitz-type q-tangent numbers --- starlike functions --- Riemann-Stieltjes functional integral --- hash --- K-functional --- (p --- Euler --- truncated-exponential polynomials --- Maple graphs --- Hurwitz-Euler eta function --- higher order Schwarzian derivatives --- generating functions --- strongly convex functions --- Hölder condition --- multiple Hurwitz-Euler eta function --- recurrence relations --- q-starlike functions --- partial sum --- Euler and Genocchi polynomials --- tangent numbers --- spectral decomposition --- determinant definition --- monomiality principle --- highly oscillatory --- Hurwitz-Lerch zeta function --- Adomian decomposition method --- analytic number theory --- existence --- existence of at least one solution --- symmetric identities --- modulus of continuity --- modified Kudryashov method --- MFCC --- q-hypergeometric functions --- differential subordination --- Janowski functions --- and Genocchi numbers --- series representation --- initial conditions --- generalization of exponential function --- upper bound --- q-derivative (or q-difference) operator --- DCT --- Schwartz testing function space --- anuran calls --- generalized Kuramoto–Sivashinsky equation --- Mittag–Leffler function --- subordination --- Hardy space --- convergence --- Hermite interpolation --- direct Hermite collocation method --- q-Euler numbers and polynomials --- distribution space --- Apostol-type polynomials and Apostol-type numbers --- Schauder fixed point theorem --- fractional integral --- convolution quadrature rule --- q)-integers --- Liouville-Caputo fractional derivative --- fixed point --- convex functions --- Grandi curves --- tempered distributions --- higher order q-Euler numbers and polynomials --- radius estimate