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1: 5.3 Graphics
See accompanying text
Figure 5.3.2: ln Γ ( x ) . This function is convex on ( 0 , ) ; compare §5.5(iv). Magnify
2: 5.5 Functional Relations
§5.5(iv) Bohr–Mollerup Theorem
If a positive function f ( x ) on ( 0 , ) satisfies f ( x + 1 ) = x f ( x ) , f ( 1 ) = 1 , and ln f ( x ) is convex (see §1.4(viii)), then f ( x ) = Γ ( x ) .
3: 1.4 Calculus of One Variable
§1.4(viii) Convex Functions
A function f ( x ) is convex on ( a , b ) if … If f ( x ) is twice differentiable, then f ( x ) is convex iff f ′′ ( x ) 0 on ( a , b ) . A continuously differentiable function is convex iff the curve does not lie below its tangent at any point.
See accompanying text
Figure 1.4.2: Convex function f ( x ) . … Magnify
4: 1.7 Inequalities
For f integrable on [ 0 , 1 ] , a < f ( x ) < b , and ϕ convex on ( a , b ) 1.4(viii)), …
5: 5.18 q -Gamma and q -Beta Functions
Also, ln Γ q ( x ) is convex for x > 0 , and the analog of the Bohr–Mollerup theorem (§5.5(iv)) holds. …
6: 20.3 Graphics
See accompanying text
Figure 20.3.2: θ 1 ( π x , q ) , 0 x 2 , q = 0. …For q q Dedekind , θ 1 ( π x , q ) is convex in x for 0 < x < 1 . … Magnify
7: Bibliography L
  • J. T. Lewis and M. E. Muldoon (1977) Monotonicity and convexity properties of zeros of Bessel functions. SIAM J. Math. Anal. 8 (1), pp. 171–178.
  • 8: Bibliography M
  • A. Michaeli (1996) Asymptotic analysis of edge-excited currents on a convex face of a perfectly conducting wedge under overlapping penumbra region conditions. IEEE Trans. Antennas and Propagation 44 (1), pp. 97–101.