scholarly journals Absolutely monotonic functions involving the complete elliptic integrals of the first kind with applications

2021 ◽  
pp. 1299-1310
Author(s):  
Zhen-Hang Yang ◽  
Jing-Feng Tian
Keyword(s):  
Elliptic Integrals ◽  
Monotonic Functions ◽  
Absolutely Monotonic Functions ◽  
Complete Elliptic Integrals ◽  
Absolutely Monotonic

scholarly journals Answers to three conjectures on convexity of three functions involving complete elliptic integrals of the first kind

2020 ◽  
Vol 14 (1) ◽  
pp. 255-271 ◽  
Author(s):  
Miao-Kun Wang ◽  
Hong-Hu Chu ◽  
Yong-Min Li ◽  
Yu-Ming Chu
Keyword(s):  
Elliptic Integral ◽  
Elliptic Integrals ◽  
Complete Elliptic Integral ◽  
Complete Elliptic Integrals ◽  
Absolutely Monotonic

In the article, we prove that the function x ? (1-x)pK(?x) is logarithmically concave on (0,1) if and only if p ? 7/32, the function x ? K(?x)/log(1+4/?1-x) is convex on (0,1) and the function x ? d2/dx2 [K(?x)- log (1+4/?1-x) is absolutely monotonic on (0,1), where K(x) = ??/20 (1-x2 sin2t)-1/2 dt (0 < x < 1) is the complete elliptic integral of the first kind.

An extension of the Rao-Rubin characterization of the Poisson distribution

1977 ◽  
Vol 14 (03) ◽  
pp. 640-646 ◽  
Author(s):  
D. N. Shanbhag
Keyword(s):  
Poisson Distribution ◽  
Present Note ◽  
Related Result ◽  
Renewal Theory ◽  
Extended Version ◽  
Monotonic Functions ◽  
Absolutely Monotonic Functions ◽  
Absolutely Monotonic

Using Bernstein's theorem concerning absolutely monotonic functions, Rao and Rubin (1964) have established a characterization of the Poisson distribution. The present note arrives at an extended version of this result using a technique existing in the renewal theory. A variant of a related result due to Srivastava and Srivastava (1970) and an extension of the result of Talwalker (1970) are also presented.

An extension of the Rao-Rubin characterization of the Poisson distribution

1977 ◽  
Vol 14 (3) ◽  
pp. 640-646 ◽  
Author(s):  
D. N. Shanbhag
Keyword(s):  
Poisson Distribution ◽  
Present Note ◽  
Related Result ◽  
Renewal Theory ◽  
Extended Version ◽  
Monotonic Functions ◽  
Absolutely Monotonic Functions ◽  
Absolutely Monotonic

Using Bernstein's theorem concerning absolutely monotonic functions, Rao and Rubin (1964) have established a characterization of the Poisson distribution. The present note arrives at an extended version of this result using a technique existing in the renewal theory. A variant of a related result due to Srivastava and Srivastava (1970) and an extension of the result of Talwalker (1970) are also presented.

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