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$$\Gamma(\alpha k+1)t^{\beta k}-\Gamma(\beta k+1)t^{\alpha k}>0$$

where $0<\alpha<\beta,$ $0<k,t$, $\Gamma(z)$ is a gamma function, $\int_0^\infty t^{z-1}e^{-t}dt$, $Re$ $z>0$.

I don't know even how to approach it. any hint, advise, counterexample, or solution would be appreciated.

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  • $\begingroup$ Is it a conjecture of yours? $\endgroup$ – Vincenzo Oliva Aug 7 '15 at 6:11
  • $\begingroup$ @VincenzoOliva yes. it is needed for my research, though unsolvable for me. :p $\endgroup$ – willbegood Aug 7 '15 at 6:14
  • $\begingroup$ Ohoh, I see. :) $\endgroup$ – Vincenzo Oliva Aug 7 '15 at 6:19
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Set $t=1$, then you easily find many counter examples because $\Gamma(x)$ is increasing for $x>1.462$. E.g. with $\alpha=2,\,\beta=3,\,k=1,\,t=1$ $$\Gamma(\alpha k+1)t^{\beta k}-\Gamma(\beta k+1)t^{\alpha k} =\Gamma(3)-\Gamma(4)=2-6=-4$$

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