*(English)*Zbl 0921.34035

The author extends in various directions the classical Lyapunov inequality for solutions to the second-order differential equation

This investigation attracted attention in several recent papers [see e.g. *B. Harris* and *Q. Kong*, Trans. Am. Math. Soc. 347, No. 5, 1831-1839 (1995; Zbl 0829.34025) and *S. Clark* and *D. B. Hinton*, Math. Inequal. Appl. 1, No. 2, 201-209 (1998; Zbl 0909.24033) and the reference given therein].

The principal role plays the concept of the downswing of a function, which measures (in a certain sense) how much a function can fall down in a given interval. Using this concept, several necessary conditions are obtained for the existence of conjugate/focal points of solutions to (*) in a given interval.

Using these results, the following interesting nonoscillation criterion for (*) is proved. If

then (*) is nonoscillatory.