New Sharp Bounds for the Modified Bessel Function of the First Kind and Toader-Qi Mean

Let <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>I</mi> <mi>v</mi> </msub> <mfenced open="(" close=")"> <mi>x</mi> </mfenced> </mrow> </semantics> </math> </inline-formula> be he modified Bessel function of the first kind of order <i>v</i>. We prove the double inequality <inline-formula> <math display="inline"> <semantics> <mrow> <msqrt> <mrow> <mfrac> <mrow> <mo form="prefix">sinh</mo> <mi>t</mi> </mrow> <mi>t</mi> </mfrac> <msup> <mo form="prefix">cosh</mo> <mrow> <mn>1</mn> <mo>/</mo> <mi>q</mi> </mrow> </msup> <mfenced separators="" open="(" close=")"> <mi>q</mi> <mi>t</mi> </mfenced> </mrow> </msqrt> <mo><</mo> <msub> <mi>I</mi> <mn>0</mn> </msub> <mfenced open="(" close=")"> <mi>t</mi> </mfenced> <mo><</mo> <msqrt> <mrow> <mfrac> <mrow> <mo form="prefix">sinh</mo> <mi>t</mi> </mrow> <mi>t</mi> </mfrac> <msup> <mo form="prefix">cosh</mo> <mrow> <mn>1</mn> <mo>/</mo> <mi>p</mi> </mrow> </msup> <mfenced separators="" open="(" close=")"> <mi>p</mi> <mi>t</mi> </mfenced> </mrow> </msqrt> </mrow> </semantics> </math> </inline-formula> holds for <inline-formula> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>></mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula> if and only if <inline-formula> <math display="inline"> <semantics> <mrow> <mi>p</mi> <mo>≥</mo> <mn>2</mn> <mo>/</mo> <mn>3</mn> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>≤</mo> <mfenced separators="" open="(" close=")"> <mo form="prefix">ln</mo> <mn>2</mn> </mfenced> <mo>/</mo> <mo form="prefix">ln</mo> <mi>π</mi> </mrow> </semantics> </math> </inline-formula>. The corresponding inequalities for means improve already known results.