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On some inequalities for the gamma and psi functions
  • H. Alzer
  • Mathematics, Computer Science
  • Math. Comput.
  • 1997
We present new inequalities for the gamma and psi functions, and we provide new classes of completely monotonic, star-shaped, and superadditive functions which are related to Γ and?.
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On some inequalities for the incomplete gamma function
  • H. Alzer
  • Mathematics, Computer Science
  • Math. Comput.
  • 1 April 1997
Let p ¬= 1 be a positive real number. We determine all real numbers α = α(p) and β = β(p) such that the inequalities formula math. formula math. are valid for all x > 0. And, we determine all realExpand
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Monotonicity theorems and inequalities for the complete elliptic integrals
We prove monotonicity properties of certain combinations of complete elliptic integrals of the first and second kind, K and E. These results lead to sharp symmetrical bounds for K and E, whichExpand
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Inequalities for the gamma function
We prove the following two theorems: (i) Let Mr(a, b) be the rth power mean of a and b. The inequality Mr(Γ(x), Γ(1/x)) ≥ 1 holds for all x ∈ (0,∞) if and only if r ≥ 1/C − π2/(6C2), where C denotesExpand
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Inequalities for Means in Two Variables
Abstract. We present various new inequalities involving the logarithmic mean $ L(x,y)=(x-y)/(\log{x}-\log{y}) $, the identric mean $ I(x,y)=(1/e)(x^x/y^y)^{1/(x-y)} $, and the classical arithmeticExpand
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Sharp upper and lower bounds for the gamma function
We prove that for all x > 0, we have with the best possible constants α = 0 and .
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Monotonicity properties of the gamma function
TLDR
We prove that G a is completely monotonic on ( 0 , ∞ ) if and only if a ≥ 1 / 3 . Expand
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Young-type inequalities and their matrix analogues
We present several new Young-type inequalities for positive real numbers and we apply our results to obtain the matrix analogues. Among others, for real numbers , and , with and , we prove theExpand
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Some classes of completely monotonic functions
We prove: (i) Let F n (r) = P n (x)[e - (1 + 1/x) x ] and G n (x) = P n (x)[(1 + 1/x) x + 1 - e], where P n (x) = x n + Σ n - 1 v = 0 c v , x v is a polynomial of degree n > 1 with real coefficients.Expand
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Inequalities for the Gamma and Polygamma Functions
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