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5 Gamma FunctionProperties

§5.6 Inequalities

Contents

§5.6(i) Real Variables

Throughout this subsection x>0.

5.6.1 1<(2π)-1/2x(1/2)-xxΓ(x)<1/(12x),
5.6.2 1Γ(x)+1Γ(1/x)2,
5.6.3 1(Γ(x))2+1(Γ(1/x))22,

Gautschi’s Inequality

5.6.4 x1-s<Γ(x+1)Γ(x+s)<(x+1)1-s,
0<s<1.

Kershaw’s Inequality

5.6.5 exp((1-s)ψ(x+s1/2))Γ(x+1)Γ(x+s)exp((1-s)ψ(x+12(s+1))),
0<s<1.

For further results see Alzer (2008), Qi (2008), Koumandos and Lamprecht (2010), and Mortici (2011b).

§5.6(ii) Complex Variables

5.6.6 |Γ(x+y)||Γ(x)|,
5.6.7 |Γ(x+y)|(sech(πy))1/2Γ(x),
x12.

For b-a1, a0, and z=x+y with x>0,

5.6.8 |Γ(z+a)Γ(z+b)|1|z|b-a.

For x0,

5.6.9 |Γ(z)|(2π)1/2|z|x-(1/2)-π|y|/2exp(16|z|-1).