Sunday, 17 April 2016

Is the function fracoperatornameHarmonic(x)operatornameAiry(x) a convex function for real numbers $0



For real numbers 0x1, I know how to exlore some aspects of the graph ( I was playing with Wolfram Alpha online calculator about its plot, see next paragraph) of the function f(x)=HxAi(x),

where Ai(x) is the Airy function, see this MathWorld and Hx is the harmonic number.




The code in Wolfram Language is



plot Harmonic(x)/Airy(x), for 0



But I don't know if it is possible to prove that this function is convex $f''(x)\geq 0$ for $0Wolfram Alpha online calculator with this code



d^2/dx^2 Harmonic(x)/Airy(x)





Question. Is it possible to prove that $$\frac{H_x}{\operatorname{Ai}(x)}\tag{1}$$ is a convex function for real numbers in the open interval $(0,1)$? Many thanks.



Answer



Mathematica gives
$$f''(0)=\frac{\pi \left(\pi \Gamma \left(\frac{2}{3}\right)^2-4 \sqrt[6]{3} \zeta (3)\right)}{\Gamma \left(\frac{1}{3}\right)} \approx -0.0162334,$$
and continuity implies that the second derivative remains negative near $x=0$. Thus $f$ is not convex over the interval $(0,1)$.


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