Generalizations Of Jordan's Inequality And Applications

In this thesis, Jordan's inequality for x∈(0,Ï€/2], see [D.S. Mitrinovic, Analytic Inequalities, Springer-Verlag,1970, P33.], is strengthened, generalized and applied.1. By using mathematical induction and L'Hospital type rules for monotonicity, Jordan's inequality is strengthened and generalized as for x∈(0,Ï€/2] and n∈N, where the coefficientsακandβκwhich are determined by recursion formulas are the best possible.Inequality (1) is applied to refine L. Yang's inequality and to estimate Sine integral function more accurately.2. The coefficients ak andακare estimated, the property of inequality (1) as nâ†'∞is investigated.3. By introducing two parametersθand t, a more general inequality for n∈N,0<x≤θ≤πand t> 2 is presented, where the constant coefficientsμκandωκwhich are determined by recursion formulas are the best possible. Inequality (2) generalizes inequality (1). Using inequality (2), L. Yang's inequality is further generalized.4. Further, a new Jordan type inequality for Bessel function is established:If k≥1/2 and 0<c<1, then is valid for any given x∈(0,Ï€/2], where the constant coefficientsγi andηi which are determined byλp(x) are the best possible, and for x∈R and real parameters c, p and b.5. Finally, inequality (3) is generalized as follows:Ifn∈N and 0<c<1, then holds for arbitrary 0<x≤θ≤π/2, where the constant coefficientsσi andνi which are determined byλp(x) are the best possible.