convexity

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1: 5.3 Graphics

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Figure 5.3.2: $\ln \mathrm{\Gamma }\left(x\right)$. This function is convex on $(0,\mathrm{\infty })$; compare §5.5(iv). Magnify

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2: 5.5 Functional Relations

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§5.5(iv) Bohr–Mollerup Theorem

►If a positive function $f(x)$ on $(0,\mathrm{\infty })$ satisfies $f(x+1)=xf(x)$, $f(1)=1$, and $\ln f(x)$ is convex (see §1.4(viii)), then $f(x)=\mathrm{\Gamma }\left(x\right)$.

3: 1.4 Calculus of One Variable

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§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 $f^{\prime \prime }(x)\ge 0$ on $(a,b)$. A continuously differentiable function is convex iff the curve does not lie below its tangent at any point. ►

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Figure 1.4.2: Convex function $f(x)$. … Magnify

4: 1.7 Inequalities

… ►For $f$ integrable on $[0,1]$, , and $\varphi$ convex on $(a,b)$ (§1.4(viii)), …

5: 5.18 $q$-Gamma and $q$-Beta Functions

… ►Also, $\ln {\mathrm{\Gamma }}_q\left(x\right)$ is convex for $x>0$, and the analog of the Bohr–Mollerup theorem (§5.5(iv)) holds. …

6: 20.3 Graphics

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Figure 20.3.2: ${\theta }_1(\pi x,q)$, $0\le x\le 2$, $q$ = 0. …For $q\le q^{\text{Dedekind}}$, ${\theta }_1(\pi x,q)$ is convex in $x$ for . … Magnify

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7: Bibliography L

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Monotonicity and convexity properties of zeros of Bessel functions. SIAM J. Math. Anal. 8 (1), pp. 171–178.

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External Links:

ISSN 1095-7154, Document, MathReview (T. Erber), ZentralBlatt

Referenced by:

§10.21(iv).

Encodings:

BibTeX

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8: Bibliography M

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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.

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External Links:

Document

Referenced by:

§9.14.

Encodings:

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