Bound/inequality for Hankel function

Solution 1:

Assume that $1\leq p \leq z$. By Nicholson’s Integral , $$ \left| {H_p^{(1)} (z)} \right|^2 = J_p^2 (z) + Y_p^2 (z) = \frac{8}{{\pi ^2 }}\int_0^{ + \infty } {\cosh (2pt)K_0 (2z\sinh t)dt} . $$ Differentiation with respect to $p$ shows that the modulus square is monotonically increasing with respect to $p$. Consequently, $\left| {H_p^{(1)} (z)} \right| \leq \left| {H_z^{(1)} (z)} \right| $ whenever $1\leq p \leq z$. By a results of this paper, $$ \left| {H_z^{(1)} (z)} \right| \!\le\! \frac{2}{{3\pi }}\frac{{\sqrt 3 }}{2}\left( {6^{1/3} \frac{{\Gamma (1/3)}}{{z^{1/3} }} + \frac{3}{{10}}\frac{1}{z}} \right) \!\le\! \frac{2}{{3\pi }}\frac{{\sqrt 3 }}{2}\left( {6^{1/3} \Gamma \!\left( {\frac{1}{3}} \right) + \frac{3}{{10}}} \right) \!=0.9497\ldots\!<1, $$ provided $z\geq 1$.