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1
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Some conjectures on the zeros of approximates to the Riemann ≡-function and incomplete gamma functions

100%
Haglund J.
Open Mathematics
|
2011
|
tom 9
|
nr 2
302-318
EN
Riemann conjectured that all the zeros of the Riemann ≡-function are real, which is now known as the Riemann Hypothesis (RH). In this article we introduce the study of the zeros of the truncated sums ≡N(z) in Riemann’s uniformly convergent infinite series expansion of ≡(z) involving incomplete gamma functions. We conjecture that when the zeros of ≡N(z) in the first quadrant of the complex plane are listed by increasing real part, their imaginary parts are monotone nondecreasing. We show how this conjecture implies the RH, and discuss some computational evidence for this and other related conjectures.
2
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On the more general Hermite-Hadamard type inequalities for convex functions via conformable fractional integrals

64%
Set E., Mumcu I., Özdemir M. E.
Topological Algebra and its Applications
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2017
|
tom 5
|
nr 1
67-73
3
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Hermite-Hadamard Type Inequalities for convex functions via generalized fractional integral operators

64%
Set E., Gözpinar A.
Topological Algebra and its Applications
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2017
|
tom 5
|
nr 1
55-62
EN
In this present work, the authors establish a new integral identity involving generalized fractional integral operators and by using this fractional-type integral identity, obtain some new Hermite-Hadamard type inequalities for functions whose first derivatives in absolute value are convex. Relevant connections of the results presented here with those earlier ones are also pointed out.
4
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Extended Riemann-Liouville type fractional derivative operator with applications

52%
Agarwal P., Nieto J. J., Luo M.-J.
Open Mathematics
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2017
|
tom 15
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nr 1
1667-1681
EN
The main purpose of this paper is to introduce a class of new extended forms of the beta function, Gauss hypergeometric function and Appell-Lauricella hypergeometric functions by means of the modified Bessel function of the third kind. Some typical generating relations for these extended hypergeometric functions are obtained by defining the extension of the Riemann-Liouville fractional derivative operator. Their connections with elementary functions and Fox’s H-function are also presented.
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