# convexity

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## 8 matching pages

##### 1: 5.3 Graphics 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 $(0,\infty)$ satisfies $f(x+1)=xf(x)$, $f(1)=1$, and $\ln f(x)$ is convex (see §1.4(viii)), then $f(x)=\Gamma\left(x\right)$.
##### 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 $f^{\prime\prime}(x)\geq 0$ on $(a,b)$. A continuously differentiable function is convex iff the curve does not lie below its tangent at any point. Figure 1.4.2: Convex function f ⁡ ( x ) . … Magnify
##### 4: 1.7 Inequalities
For $f$ integrable on $[0,1]$, $a, and $\phi$ convex on $(a,b)$1.4(viii)), …
##### 5: 5.18 $q$-Gamma and $q$-Beta Functions
Also, $\ln\Gamma_{q}\left(x\right)$ is convex for $x>0$, and the analog of the Bohr–Mollerup theorem (§5.5(iv)) holds. …
##### 6: 20.3 Graphics 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.