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An extension of Vietoris's inequalities
We establish a best possible extension of a famous Theorem of Vietoris about the positivity of a general class of cosine sums. Our result refines and sharpens several earlier generalizations of thisExpand
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Remarks on some completely monotonic functions
Applying the Euler–Maclaurin summation formula, we obtain upper and lower polynomial bounds for the function xex−1, x>0, with coefficients the Bernoulli numbers Bk. This enables us to give simplerExpand
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On a conjecture for trigonometric sums and starlike functions
We prove the case @r=14 of the following conjecture of Koumandos and Ruscheweyh: let @m^*(@r) be the unique solution of @!"0^(^@r^+^1^)^@psin([email protected]@p)t^@m^-^1dt=0 in (0,1]. Expand
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Inequalities for Trigonometric Sums
We give a survey of recent results on positive trigonometric sums. Far-reaching extensions and generalizations of classical results are presented. We provide new proofs as well as additional remarksExpand
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On a monotonic trigonometric sum
By establishing a cosine analogue of a result of Askey and Steinig on a monotonic sine sum, this paper sharpens and unifies several results associated with Young's inequality for the partial sums ofExpand
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Completely monotonic functions of positive order and asymptotic expansions of the logarithm of Barnes double gamma function and Euler's gamma function
Abstract We introduce completely monotonic functions of order r > 0 and show that the remainders in asymptotic expansions of the logarithm of Barnes double gamma function and Euler's gamma functionExpand
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On a conjecture of Clark and Ismail
Let Φm(x) = -xm ψ(m) (x),where ψ denotes the logarithmic derivative of Euler's gamma function. Clark and Ismail prove in a recently published article that if m ∈ {1, 2 ..... 16}, then Φm(m) isExpand
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The zeros of certain Lommel functions
Lommel’s function sμ,ν(z) is a particular solution of the differential equation z2y′′ + zy′ + (z2 − ν2)y = zμ+1. Here we present estimates and monotonicity properties of the positive zeros ofExpand
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On the Positivity of Some Basic Legendre Polynomial Sums
The positivity of the partial sums of the series of Legendre polynomials is a classical result of Fejer. The paper presents corresponding results for both even and odd Legendre polynomials with bestExpand
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Series representations for γ and other mathematical constants
AbstractWe present series representations for some mathematical constants, like γ, π, log 2, ζ(3). In particular, we prove that the following representation for Euler’s constant is valid: $$ \gamma =Expand
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