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On the Inequality of P. Turan for Legendre Polynomials  
 
  Authors: Eugen Constantinescu,  
  Keywords: Orthogonal polynomials, Legendre polynomials, Turan Inequality, Positivity.  
  Date Received: 10/01/05  
  Date Accepted: 03/02/05  
  Subject Codes:

33C10, 26D20.

  
  Editors: Alexandru Lupas (1942-2007),  
 
  Abstract:

Our aim is to prove the inequalities

begin{displaymath} frac{1-x^{2}}{n(n+1)}h_{n}leq begin{array}{vert llvert}... ...{1-x^{2}}{2};,quad forall xin lbrack -1,1],;n=1,2,dots, end{displaymath}

where $ h_{n}:=sum_{k=1}^{n}frac{1}{k}$ and $ left( P_{n}right) _{n=0}^{infty }$ are the Legendre polynomials . At the same time, it is shown that the sequence having as general term

begin{displaymath}n(n+1) begin{array}{vert llvert} P_{n}(x) & P_{n+1}(x) [3pt] P_{n-1}(x) & P_{n}(x) end{array}end{displaymath}

is non-decreasing for $ xin lbrack -1,1].$ ;


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