with the best possible constants
This refines a recently published result of Batir, who proved that the double-inequality holds with and b=γ.  相似文献
5.
Remark on a double-inequality for the Riemann zeta function     
Horst Alzer   《Expositiones Mathematicae》2005,23(4):349-352
Let ζ be the Riemann zeta function and δ(x)=1/(2x-1). For all x>0 we have
(1-δ(x))ζ(x)+αδ(x)<ζ(x+1)<(1-δ(x))ζ(x)+βδ(x),
with the best possible constant factors
This improves a recently published result of Cerone et al., J. Inequalities Pure Appl. Math. 5(2) (43) (2004), who showed that the double-inequality holds with and .  相似文献
6.
Inequalities of Fejér-Jackson Type     
Horst Alzer  Stamatis Koumandos 《Monatshefte für Mathematik》2003,139(2):89-103
 We complement, extend, and sharpen some known inequalities for sine sums. Our main result is the following refinement of the classical Fejér-Jackson inequality: For all integers n ⩾ 2 and real numbers x ∈ (0,π) we have
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1.
On some inequalities for the gamma and psi functions   总被引:12,自引:0,他引:12  
We present new inequalities for the gamma and psi functions, and we provide new classes of completely monotonic, star-shaped, and super-additive functions which are related to and .

  相似文献

2.
On some inequalities for the incomplete gamma function   总被引:4,自引:0,他引:4  
Let be a positive real number. We determine all real numbers and such that the inequalities

are valid for all . And, we determine all real numbers and such that

hold for all .

  相似文献

3.
Let and be real numbers. The inequality

holds for all positive real numbers if and only if . The reverse inequality is valid for all if and only if .

  相似文献

4.
Let Hn be the nth harmonic number. For all integers n1 we have
a-log(e1/(n+1)-1)Hn<b-log(e1/(n+1)-1),
with the best possible constant factor α = 1. This improves an inequality due to Turán. Received February 12, 2002 Published online April 4, 2003  相似文献
7.
Inequalities for Cyclic Functions     
Horst Alzer  Stephan Ruscheweyh  Luis Salinas 《Journal of Approximation Theory》2001,112(2):216
The nth cyclic function is defined by

We prove that if k is an integer with 1kn−1, then

holds for all positive real numbers x with the best possible constantsα=1 and β= 2n-k over n.  相似文献
8.
On the intersection of two-parameter mean value families     
Horst Alzer  Stephan Ruscheweyh 《Proceedings of the American Mathematical Society》2001,129(9):2655-2662

We determine all means which are in the intersection of two multivariable two-parameter mean value families. These families were introduced by C. Gini (1938) and K.B. Stolarsky (1975).

  相似文献

9.
Inequalities for the gamma function     
Horst Alzer 《Proceedings of the American Mathematical Society》2000,128(1):141-147
We prove the following two theorems:

(i) Let be the th power mean of and . The inequality

holds for all if and only if , where denotes Euler's constant. This refines results established by W. Gautschi (1974) and the author (1997).

(ii) The inequalities

are valid for all if and only if and , while holds for all if and only if and . These bounds for improve those given by G. D. Anderson an S.-L. Qiu (1997).

  相似文献

10.
A Refinement of Carleman's Inequality     
Horst Alzer 《Journal of Approximation Theory》1998,95(3):497-499
We prove a new sharpening of the inequalitywhich is due to T. Carleman.  相似文献
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