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Let $0 < \delta < 1$, and let $I_\delta$ be the set of all complex numbers $\mu$ such that $-1/2 + \delta < \Re \mu < 1/2 -\delta$. Is there a polynomial $P_\delta$ such that for all $\mu \in I_\delta$, $$ \sup_{y > 0} y^{1/2} \frac{|K_\mu(2 \pi y)|}{|K_\mu(2 \pi)|} < P_\delta(\Im \mu) $$ with $K_\mu$ the $K$-Bessel function?

In general, are there any reasonable lower bounds on $K$-Bessel functions of non-real order?

We need this in a work on the distribution of rational points on a class of algebraic varieties.

Thanks!

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    $\begingroup$ If you don't mind having $\mu$ purely imaginary, then Balogh's uniform asymptotic expansion of the Bessel function (paper from mid 1960's) should give you the truth. This may be overkill. $\endgroup$
    – Matt Young
    Dec 7, 2015 at 22:04
  • $\begingroup$ @Matt: Oh I wish! $\mu$ being purely imaginary is Ramnujan's conjecture. $\endgroup$
    – Ramin
    Dec 7, 2015 at 23:24
  • $\begingroup$ That's unfortunate. I searched on mathscinet for the papers that referenced Balogh's, and I didn't see anything that applied for more general $\mu$. My gut feeling is that varying the real part of $\mu$ slightly should be possible with current technology, however. Do you care very much about the degree of the polynomial? $\endgroup$
    – Matt Young
    Dec 8, 2015 at 15:25
  • $\begingroup$ No, the degree of the polynomial is ok. We did find two papers though: 1. ASYMPTOTICS OF MODIFIED BESSEL FUNCTIONS OF HIGH ORDER by Sidi and Hoggan (2011), and 2. ASYMPTOTIC EXPANSIONS OF MELLIN TRANSFORMS AND ANALOGUES OF WATSON’S LEMMA by Sidi (1985) which might be sufficient for our purposes. $\endgroup$
    – Ramin
    Dec 8, 2015 at 19:48

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