# convex functions

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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: 1.4 Calculus of One Variable
###### §1.4(viii) ConvexFunctions
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. Figure 1.4.2: Convex function f ⁡ ( x ) . … Magnify
##### 3: 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 $f(x)=\Gamma\left(x\right)$.
##### 4: 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.
• ##### 5: 1.7 Inequalities
For $f$ integrable on $[0,1]$, $a, and $\phi$ convex on $(a,b)$1.4(viii)), …
##### 6: 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. …
##### 7: Bibliography M
• I. G. Macdonald (1990) Hypergeometric Functions.
• B. Markman (1965) Contribution no. 14. The Riemann zeta function. BIT 5, pp. 138–141.
• F. Matta and A. Reichel (1971) Uniform computation of the error function and other related functions. Math. Comp. 25 (114), pp. 339–344.
• 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.
• S. C. Milne (2002) Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions. Ramanujan J. 6 (1), pp. 7–149.
• ##### 8: 20.3 Graphics
###### §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. …