Abstract
Let Ipx be the modified Bessel function of the first kind of order p. The upper and lower bounds in the form of simple rational functions about cosht and (sinht)/t for the function I0x are obtained. The corresponding inequalities for the Toader-Qi mean do not match those in the existing literature.
Highlights
We know that the modified Bessel function of the first kind of order p is denoted as
X2 y00 ( x ) + xy0 ( x ) − x2 + p2 y( x ) = 0, which can be written as the infinite series the Modified Bessel Function of the
(i) The numerical results show that inequality (20) is sharper than the one (38) on
Summary
We know that the modified Bessel function of the first kind of order p is denoted as. We first consider finding the bound of TQ( a, b) in the following form: x1 A2 + y1 AL + z1 G2. A + w1 G which is equivalent to searching for a bound for TQ( a, b)/G = I0 (t) in the form of ( A + w1 G ) G In this way, we can consider the power series expansion h i (cosh t + w1 ) I0 (t) − x1 (cosh t)2 + y1 (cosh t)((sinh t)/t) + z1.
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