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Two-sided bounds uniform in the real argument and the index for modified Bessel functions. (English. Russian original) Zbl 0952.33002
Two-sided bounds are derived for the modified Bessel functions and the functions $a_\nu(x)=xI_\nu'(x)/I_\nu(x)$ and $b_\nu(x)=xK_\nu'(x)/K_\nu(x)$ for $x>0$, $\nu\ge 0$, except for some neighborhoods of the point $(x,\nu)=(0,0)$. The bounds are obtained by using the Riccati equation for $a_\nu(x)$, $b_\nu(x)$, and a general theorem on inequalities for solutions of a type of differential equations.

##### MSC:
 33C10 Bessel and Airy functions, cylinder functions, ${}_0F_1$ 34C11 Qualitative theory of solutions of ODE: growth, boundedness 26D07 Inequalities involving other types of real functions 26D10 Inequalities involving derivatives, differential and integral operators
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##### References:
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