On certain comparison theorems for second order linear oscillation

Abstract

It is shown that if ( p y ′ ) ′ + q y = 0 (py’)’ + qy = 0 on [ 0 , ∞ ) [0,\infty ) is oscillatory then ( p z ′ ) ′ + a q z = 0 (pz’)’ + aqz = 0 is also oscillatory for functions satisfying a ( t ) ⩾ 1 a(t) \geqslant 1 and \[ 2 p ( t ) a ′ ( t ) − 3 ∫ 0 t p ( s ) a ′ 2 ( s ) a − 1 ( s ) d s 2p(t)a’(t) - 3\int _0^t {p(s){{a’}^2}(s){a^{ - 1}}(s)ds} \] is nonincreasing.