I need a special inequality related to a fractional derivative problem. Let k∈ℕ ,0<α<1 , 0<β<1.Consider : A=[Γ(1-α)Γ(1+k-β)/Γ(2-β-α+k)].(1-α) On what conditions (on k ,β and α) A is less than 1. I do not know anything about this kind of questions. Thank's a lot.
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put on hold as off-topic by Stopple, Steven Sam, Stefan Kohl, Lucia, abx 9 hours agoThis question appears to be off-topic. The users who voted to close gave this specific reason:
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Stopple is right, $A=B(1-\alpha, 1+k-\beta)$, where $B$ is the Beta function. However it seems in your formulation that $\alpha,\beta$ are fixed and $k$ is running through $\mathbb N$. Then a simple application of Stirling's formula gives $$ B(1-\alpha, 1+k-\beta)\sim (1+k+\beta)^{\alpha-1}\Gamma(1-\alpha),\quad k\rightarrow +\infty. $$ In particular, it goes to zero when $k$ goes to infinity. |
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