Let A be a class of functions f(z) of the form f(z) = z + Sigma(infinity)(n=2) a(n)z(n) (0.1) which are analytic in the open unit disk U. By means of the Dziok-Srivastava operator, we introduce a new subclass S'(m...
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Let A be a class of functions f(z) of the form f(z) = z + Sigma(infinity)(n=2) a(n)z(n) (0.1) which are analytic in the open unit disk U. By means of the Dziok-Srivastava operator, we introduce a new subclass S'(m) (alpha(1), alpha, mu) (l <= m + 1, l, m is an element of N boolean OR uniformly convex functions, -pi/2 < alpha < pi/2, mu > - cos alpha) of A. In particular, S-0(1) (2, 0, 0) coincides with the class of uniformly convex functions introduced by Goodman. The order of starlikeness and the radius of alpha-spirallikeness of order beta (beta < 1) are computed. Inclusion relations and convolution properties for the class S'(m) (alpha(1), alpha, mu) are obtained. A special member of S'(m) (alpha(1), alpha, mu) is also given. The results presented here not only generalize the corresponding known results, but also give rise to several other new results.
An operator I is said to be an averaging (or mean-value) operator on a set kappa of analytic functions in Delta = {z : vertical bar z vertical bar < 1}, if I [f](0) = f (0) and I [f](Delta) is contained in the conv...
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An operator I is said to be an averaging (or mean-value) operator on a set kappa of analytic functions in Delta = {z : vertical bar z vertical bar < 1}, if I [f](0) = f (0) and I [f](Delta) is contained in the convex hull of f (Delta) for all f epsilon kappa. In this work we consider the class SP(alpha) of functions defined by us (Folia Sci Univ Technol Resov 28:35-42, 1993), which is connected with the class of uniformly convex functions introduced by Goodman (Ann Polon Math 56:87-92, 1991). We describe an interesting new construction of averaging operators which might attract a considerable attention of mathematicians working in the field.
For q a (0, 1) let the q-difference operator be defined as follows where denotes the open unit disk in a complex plane. Making use of the above operator the extended Ruscheweyh differential operator R (q) (lambda) f i...
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For q a (0, 1) let the q-difference operator be defined as follows where denotes the open unit disk in a complex plane. Making use of the above operator the extended Ruscheweyh differential operator R (q) (lambda) f is defined. Applying R (q) (lambda) f a subfamily of analytic functions is defined. Several interesting properties of a defined family of functions are investigated.
We establish some conditions under which the differential subordination of the type p(z) + zp'(z)/p(z) < Q(z) yields p < q in U. functions Q and q are chosen so that they map the unit disk onto domains enclo...
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We establish some conditions under which the differential subordination of the type p(z) + zp'(z)/p(z) < Q(z) yields p < q in U. functions Q and q are chosen so that they map the unit disk onto domains enclosed by conic sections. Some applications of obtained results are given.
The coefficients of a plant of an abstract control problem are perturbed. It is shown that the characterizations given by Zolezzi for the abstract quadratic control problem in Hilbert spaces remain true for more gener...
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The coefficients of a plant of an abstract control problem are perturbed. It is shown that the characterizations given by Zolezzi for the abstract quadratic control problem in Hilbert spaces remain true for more general functions and spaces.
We study stochastic convex optimization under infinite noise variance. Specifically, when the stochastic gradient is unbiased and has uniformly bounded (1+ k)-th moment, for some k is an element of (0;1], we quantify ...
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We study stochastic convex optimization under infinite noise variance. Specifically, when the stochastic gradient is unbiased and has uniformly bounded (1+ k)-th moment, for some k is an element of (0;1], we quantify the convergence rate of the Stochastic Mirror Descent algorithm with a particular class of uniformlyconvex mirror maps, in terms of the number of iterations, dimensionality and related geometric parameters of the optimization problem. Interestingly this algorithm does not require any explicit gradient clipping or normalization, which have been extensively used in several recent empirical and theoretical works. We complement our convergence results with information-theoretic lower bounds showing that no other algorithm using only stochastic first-order oracles can achieve improved rates. Our results have several interesting consequences for devising online/streaming stochastic approximation algorithms for problems arising in robust statistics and machine learning.
We define the generalized Dziok-Srivastava operator for harmonic functions and introduce a new subclass of complex valued harmonic functions which are orientation preserving and univalent in the open unit disc. We inv...
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We define the generalized Dziok-Srivastava operator for harmonic functions and introduce a new subclass of complex valued harmonic functions which are orientation preserving and univalent in the open unit disc. We investigate the coefficient bounds, distortion inequalities and extreme points for this generalized class of functions.
The purpose of the present paper is to investigate some sufficient conditions for convolution operator H(k1,c)f (z) = zu(p)(z) * f(z) belonging to the classes k - UCV(alpha), k - S-p(a), S-lambda*, and C-lambda.
The purpose of the present paper is to investigate some sufficient conditions for convolution operator H(k1,c)f (z) = zu(p)(z) * f(z) belonging to the classes k - UCV(alpha), k - S-p(a), S-lambda*, and C-lambda.
In this paper we investigate several interesting properties of the fractional calculus operators, the Carlson-Shaffer linear operator L(a,c) and a certain integral operator, associated with various subclasses of analy...
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In this paper we investigate several interesting properties of the fractional calculus operators, the Carlson-Shaffer linear operator L(a,c) and a certain integral operator, associated with various subclasses of analytic and univalent functions. We also investigate the relationships of some of these classes with the Hardy spaceH∞(of bounded analytic functions in the open unit disk μ and with some families of multiplier transformations.
In this paper, we introduce notable Jensen-Mercer inequality for a general class of convexfunctions, namely uniformly convex functions. We explore some interesting properties of such a class of functions along with s...
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In this paper, we introduce notable Jensen-Mercer inequality for a general class of convexfunctions, namely uniformly convex functions. We explore some interesting properties of such a class of functions along with some examples. As a result, we establish Hermite-Jensen-Mercer inequalities pertaining uniformly convex functions by considering the class of fractional integral operators. Moreover, we establish Mercer-Ostrowski inequalities for conformable integral operator via differentiable uniformly convex functions. Finally, we apply our inequalities to get estimations for normal probability distributions (Gaussian distributions).
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