On estimation of the derivative of the Legendre polynomial
Estimates of the derivative of the Legendre polynomial on the orthogonality segment <i>−</i>1 <i>x </i> 1 of the type 1 <i>− x</i>2 <i>α dPn</i>(<i>x</i>)<i>/dx < A</i>(<i>n </i>+ <i>a</i>)<i>δ </i>are well known. In particular, if <i>α </i>= 0 then <i>δ </i>= 2, <i>A </i>= 1<i>/</i>2; if <i>α </i>= 1 then <i>δ </i>= 1<i>/</i>2, <i>A </i>= *2<i>/π </i>= 0<i>.</i>797885. In both cases the constants <i>δ</i>, <i>A </i>are unimprovable. An estimate <i>δ </i>= 1<i>/</i>2, <i>A </i>= *4<i>/π </i>= 1<i>.</i>128379 in case <i>α </i>= 3<i>/</i>4 occurs in literature. Here we have found exact values of main constants if <i>α </i>= 3<i>/</i>4: <i>δ </i>= 1 2 <i>, A</i>= max 0 <i>t<∞ √tJ</i>1(<i>t</i>) = 0<i>.</i>825031<i>, J</i>1 being the Bessel function. It is admissible to adopt <i>a </i>= 2<i>/</i>3 for all <i>n </i> 0. The set of <i>a</i>-values for which the inequality under consideration is true for sufficiently large <i>n </i>has the lower bound equal to 1<i>/</i>2, though the limiting value <i>a </i>= 1<i>/</i>2 is not acceptable.