On certain inequalities and characteristic value problems for analytic functions and for functions of two variables

Abstract

ANALYTIC FUNCTIONS 323 periodic function z = z(s) of a parameter s.Except at a finite number of corners this function shall have a continuous derivative ¿Tá(s)' which, at each corner, is continuous on both sides.We can assume dz z = ds We set Az(s) = Then we have (2.3)'KURT FRIEDRICHS [May z(s + a) -z(s) Az(s) = ¿(s + a) -z(s) Az(s)-z(s)\ g-, 4 3 .5 -^ Az(s) ^-4 4 We choose a number p >0 in such a way that | s' -s | ^ o-as | z(s') -z(s) | g p and a number T > 0 in such a way that and (2.4) T Consequently we have (2.4)'Az(s') -Az(s) rs7> iS -as 40 î ^ <r.á-|s'-s|+^-|5'-s( + -\t'-t\ 4 ' ' 40 ' 4 ' ' ce e -+ -+» -4 40 4 (í' -í) + *(/'-0|.Now let z' = z.Then we have | z(s') -z(s) I = I t'Az(s') -/Az(s) | ^ /' | Az(s') | + /1 Az(j) | 5 ^ 27-^ p. 4