convex functions

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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: 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 … ►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

3: 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 $f(x)=\mathrm{\Gamma }\left(x\right)$.

4: 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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5: 1.7 Inequalities

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

6: 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. …

7: Bibliography M

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Hypergeometric Functions.

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Notes:

Unpublished Lecture Notes, University of London. An online version exists.

External Links:

FullText

Referenced by:

§35.7(ii), §35.8(iii).

Encodings:

BibTeX

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Contribution no. 14. The Riemann zeta function. BIT 5, pp. 138–141.

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Referenced by:

§25.21(ii).

Encodings:

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Uniform computation of the error function and other related functions. Math. Comp. 25 (114), pp. 339–344.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§7.22(i).

Encodings:

BibTeX

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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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Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions. Ramanujan J. 6 (1), pp. 7–149.

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External Links:

Document, ZentralBlatt

Referenced by:

§27.13(iv).

Encodings:

BibTeX

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8: 20.3 Graphics

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§20.3(i) $\theta$-Functions: Real Variable and Real Nome

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§20.3(ii) $\theta$-Functions: Complex Variable and Real Nome

►In the graphics shown in this subsection, height corresponds to the absolute value of the function and color to the phase. … ►

§20.3(iii) $\theta$-Functions: Real Variable and Complex Lattice Parameter

►In the graphics shown in this subsection, height corresponds to the absolute value of the function and color to the phase. …