scholarly journals Q-Extension of Starlike Functions Subordinated with a Trigonometric Sine Function

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1676
Author(s):  
Saeed Islam ◽  
Muhammad Ghaffar Khan ◽  
Bakhtiar Ahmad ◽  
Muhammad Arif ◽  
Ronnason Chinram
Keyword(s):  
Extreme Point ◽  
Point Theorem ◽  
Starlike Functions ◽  
Partial Sums ◽  
Sufficient Condition ◽  
Sine Function ◽  
Necessary And Sufficient Condition ◽  
Closure Theorem ◽  
Necessary And Sufficient ◽  
Distortion Bound

The main purpose of this article is to examine the q-analog of starlike functions connected with a trigonometric sine function. Further, we discuss some interesting geometric properties, such as the well-known problems of Fekete-Szegö, the necessary and sufficient condition, the growth and distortion bound, closure theorem, convolution results, radii of starlikeness, extreme point theorem and the problem with partial sums for this class.

scholarly journals Controllability of Nonlinear Fractional Dynamical Systems with a Mittag–Leffler Kernel

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2139
Author(s):  
Jiale Sheng ◽  
Wei Jiang ◽  
Denghao Pang ◽  
Sen Wang
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Fixed Point Theorem ◽  
Laplace Transform ◽  
Point Theorem ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

This paper is concerned with controllability of nonlinear fractional dynamical systems with a Mittag–Leffler kernel. First, the solution of fractional dynamical systems with a Mittag–Leffler kernel is given by Laplace transform. In addition, one necessary and sufficient condition for controllability of linear fractional dynamical systems with Mittag–Leffler kernel is established. On this basis, we obtain one sufficient condition to guarantee controllability of nonlinear fractional dynamical systems with a Mittag–Leffler kernel by fixed point theorem. Finally, an example is given to illustrate the applicability of our results.

scholarly journals On q-analogue of Janowski-type starlike functions with respect to symmetric points

2021 ◽  
Vol 54 (1) ◽  
pp. 37-46
Author(s):  
Muhammad Ghaffar Khan ◽  
Bakhtiar Ahmad ◽  
Raees Khan ◽  
Muhammad Zubair ◽  
Zabidin Salleh
Keyword(s):  
Extreme Point ◽  
Point Theorem ◽  
Starlike Functions ◽  
Circular Domain ◽  
Distortion Theorem ◽  
Geometric Properties ◽  
Coefficient Bounds ◽  
Closure Theorem ◽  
Symmetric Points

Abstract The main objective of the present paper is to define a class of q q -starlike functions with respect to symmetric points in circular domain. Some interesting results of these functions have been evaluated in this article. The sufficiency criteria in the form of convolutions are evaluated. Furthermore, other geometric properties such as coefficient bounds, distortion theorem, closure theorem and extreme point theorem are also obtained for these newly defined functions.

THE VARIANCE OF AN INTEGRATED PROCESS NEED NOT DIVERGE TO INFINITY, AND RELATED RESULTS ON PARTIAL SUMS OF STATIONARY PROCESSES

2001 ◽  
Vol 17 (4) ◽  
pp. 671-685 ◽  
Author(s):  
Hannes Leeb ◽  
Benedikt M. Pötscher
Keyword(s):  
Sufficient Conditions ◽  
Partial Sums ◽  
Stationary Processes ◽  
Sufficient Condition ◽  
Integrated Process ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

For a process with stationary first differences a necessary and sufficient condition for the variance of the process to be unbounded is given. An example shows that the variance of an integrated process—although unbounded—need not diverge to infinity. Sufficient conditions for the variance of an integrated process to diverge to infinity are provided.

scholarly journals Applications of Mittag-Leffer Type Poisson Distribution to a Subclass of Analytic Functions Involving Conic-Type Regions

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Muhammad Ghaffar Khan ◽  
Bakhtiar Ahmad ◽  
Nazar Khan ◽  
Wali Khan Mashwani ◽  
Sama Arjika ◽  
...  
Keyword(s):  
Poisson Distribution ◽  
Analytic Functions ◽  
Convex Combination ◽  
Partial Sums ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Geometric Properties ◽  
Distortion Bounds ◽  
Janowski Functions ◽  
Necessary And Sufficient

In this article, we introduce a new subclass of analytic functions utilizing the idea of Mittag-Leffler type Poisson distribution associated with the Janowski functions. Further, we discuss some important geometric properties like necessary and sufficient condition, convex combination, growth and distortion bounds, Fekete-Szegö inequality, and partial sums for this newly defined class.

scholarly journals Property preserving operators

1989 ◽  
Vol 39 (3) ◽  
pp. 397-404
Author(s):  
Evelyn M. Silvia
Keyword(s):  
Hadamard Product ◽  
Starlike Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Real Numbers ◽  
Gauss Function ◽  
Necessary And Sufficient ◽  
Class Of Functions ◽  
Hypergeometric Gauss Function

Let S denote the class of functions of the form that are analytic and univalent in |z| < 1. Given f ∈ S and a, b, c, real numbers other than 0, −1, −2,…, let Ω(a, b, C; f) = F(a, b, C; z)* f(z) where is a hypergeometric Gauss function with (a)0 = 1 and (a)k = a(a + 1) … (a + k − 1) and * denotes the Hadamard product. For qn(z) = z + a2z2 + … + anzn (an ≠ 0, n = 5,6) in S, it is shown that , is univalent in |z| < 1. This extends the result previously known for n = 3 and n = 4. Also, we obtain a necessary and sufficient condition involving a, b, and c such that Ω(a, b, c;·) preserves the subclass of S consisting of starlike functions of order α, 0 ≤ α ≤ 1, with ak 0.

scholarly journals A Note on the Signature Representations of the Symmetric Groups

2021 ◽  
Vol 28 (03) ◽  
pp. 469-478
Author(s):  
Kay Jin Lim ◽  
Jialin Wang
Keyword(s):  
Symmetric Groups ◽  
Partial Sums ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Composition Formula ◽  
Kostka Numbers ◽  
Necessary And Sufficient

For a partition [Formula: see text] and a prime [Formula: see text], we prove a necessary and sufficient condition for there to exist a composition [Formula: see text] such that [Formula: see text] can be obtained from [Formula: see text] after rearrangement and no partial sums of [Formula: see text] are divisible by [Formula: see text]. To demonstrate why we are interested in the question, we compute some signed [Formula: see text]-Kostka numbers.

scholarly journals Invariance principle via orthomartingale approximation

2018 ◽  
Vol 18 (06) ◽  
pp. 1850043 ◽  
Author(s):  
Davide Giraudo
Keyword(s):  
Invariance Principle ◽  
Random Fields ◽  
Partial Sums ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Stationary Random Fields ◽  
Necessary And Sufficient

We obtain a necessary and sufficient condition for the orthomartingale-coboundary decomposition. We establish a sufficient condition for the approximation of the partial sums of a strictly stationary random fields by those of stationary orthomartingale differences. This condition can be checked under multidimensional analogues of the Hannan condition and the Maxwell–Woodroofe condition.

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