<p>We study the behavior of the smallest possible constant <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(d(a,b)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> in Hardy inequality<Equation ID="Equa"> <EquationSource Format="TEX">\(\int_a^b\Big(\frac{1}{x}\int_a^xf(t) \, dt \Big)^p\,dx\leqd(a,b)\int_a^b [f(x)]^p \,dx.\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msubsup> <mo>∫</mo> <mi>a</mi> <mi>b</mi> </msubsup> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">(</mo> </mrow> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> <msubsup> <mo>∫</mo> <mi>a</mi> <mi>x</mi> </msubsup> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="0.166667em" /> <mi>d</mi> <mi>t</mi> <msup> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">)</mo> </mrow> <mi>p</mi> </msup> <mspace width="0.166667em" /> <mi>d</mi> <mi>x</mi> <mo>≤</mo> <mi>d</mi> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <msubsup> <mo>∫</mo> <mi>a</mi> <mi>b</mi> </msubsup> <msup> <mrow> <mo stretchy="false">[</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">]</mo> </mrow> <mi>p</mi> </msup> <mspace width="0.166667em" /> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mrow> </math></EquationSource> </Equation>The exact rate of convergence of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(d(a,b)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is established and the “almost extremal” function is found.</p>