On Riesz and generalised Ces�ro summability of arbitrary positive order

1967 ◽  
Vol 99 (2) ◽  
pp. 171-177 ◽  
Author(s):  
D. Borwein ◽  
D. C. Russell
Keyword(s):  
Positive Order ◽  
Arbitrary Positive Order

scholarly journals Arbitrary Order Fractional Difference Operators with Discrete Exponential Kernels and Applications

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Thabet Abdeljawad ◽  
Qasem M. Al-Mdallal ◽  
Mohamed A. Hajji
Keyword(s):  
Type Inequality ◽  
Difference Operators ◽  
Fractional Difference ◽  
Positive Order ◽  
Contraction Theorem ◽  
Banach Contraction ◽  
Sturm Liouville ◽  
Initial Difference ◽  
Arbitrary Positive Order ◽  
Banach Contraction Theorem

Recently, Abdeljawad and Baleanu have formulated and studied the discrete versions of the fractional operators of order0<α≤1with exponential kernels initiated by Caputo-Fabrizio. In this paper, we extend the order of such fractional difference operators to arbitrary positive order. The extension is given to both left and right fractional differences and sums. Then, existence and uniqueness theorems for the Caputo (CFC) and Riemann (CFR) type initial difference value problems by using Banach contraction theorem are proved. Finally, a Lyapunov type inequality for the Riemann type fractional difference boundary value problems of order2<α≤3is proved and the ordinary difference Lyapunov inequality then follows asαtends to2from right. Illustrative examples are discussed and an application about Sturm-Liouville eigenvalue problem in the sense of this new fractional difference calculus is given.

scholarly journals On Absolute Summability for any Positive Order

1933 ◽  
Vol 3 (3) ◽  
pp. 173-178 ◽  
Author(s):  
C. E. Winn
Keyword(s):  
Absolute Summability ◽  
Positive Order ◽  
Partial Sum ◽  
Image Position

Absolute summability according to Cesàro's method has been defined by Fekete for positive integral orders, as follows:—Denoting the rth partial sum of a series Σun by and its rth mean, namely2, by we can regard as the sum of the series.

On the Points of Inflection of Bessel Functions of Positive Order, I

1990 ◽  
Vol 42 (5) ◽  
pp. 933-948 ◽  
Author(s):  
Lee Lorch ◽  
Peter Szego
Keyword(s):  
Bessel Function ◽  
Bessel Functions ◽  
Primary Concern ◽  
Second Derivative ◽  
Positive Order ◽  
Fixed Rank

The primary concern addressed here is the variation with respect to the order v > 0 of the zeros jʺvk of fixed rank of the second derivative of the Bessel function Jv(x) of the first kind. It is shown that jʺv1 increases 0 < v < ∞ (Theorem 4.1) and that jʺvk increases in 0 < v ≤ 3838 for fixed k = 2, 3,… (Theorem 10.1).

On weak convergence within the hnbue family of life distributions

1984 ◽  
Vol 21 (03) ◽  
pp. 654-660 ◽  
Author(s):  
Sujit K. Basu ◽  
Manish C. Bhattacharjee
Keyword(s):  
Weak Convergence ◽  
Exponential Distribution ◽  
Limiting Distribution ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Moment Sequence ◽  
Positive Order ◽  
Life Distributions ◽  
Necessary And Sufficient

We show that the HNBUE family of life distributions is closed under weak convergence and that weak convergence within this family is equivalent to convergence of each moment sequence of positive order to the corresponding moment of the limiting distribution. A necessary and sufficient condition for weak convergence to the exponential distribution is given, based on a new characterization of exponentials within the HNBUE family of life distributions.

SOME REMARKS ON FRACTIONAL INTEGRAL OF ONE-DIMENSIONAL CONTINUOUS FUNCTIONS

Fractals ◽  
2020 ◽  
Vol 28 (01) ◽  
pp. 2050005
Author(s):  
JIA YAO ◽  
YING CHEN ◽  
JUNQIAO LI ◽  
BIN WANG
Keyword(s):  
Continuous Function ◽  
Fractional Integral ◽  
Continuous Functions ◽  
Box Dimension ◽  
Positive Order ◽  
One Dimensional ◽  
Katugampola Fractional Integral ◽  
Hadamard Fractional Integral

In this paper, we make research on Katugampola and Hadamard fractional integral of one-dimensional continuous functions on [Formula: see text]. We proved that Katugampola fractional integral of bounded and continuous function still is bounded and continuous. Box dimension of any positive order Hadamard fractional integral of one-dimensional continuous functions is one.

scholarly journals Convex and starlike criteria

1999 ◽  
Vol 22 (1) ◽  
pp. 75-79 ◽  
Author(s):  
Herb Silverman
Keyword(s):  
Sufficient Conditions ◽  
Starlike Functions ◽  
Positive Order ◽  
Convex And Starlike Functions

We investigate an expression involving the quotient of the analytic representations of convex and starlike functions. Sufficient conditions are found for functions to be starlike of a positive order and convex.

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