Differential inequalities and Carathodory functions

The author proves a very general result from which it is possible to show that a regular function satisfying a differential inequality of a certain type is necessarily a Caratheodory function. This result has applications in the theory of univalent functions. Let 8P denote the class of Caratheodory functions; that is, functions p(z)=l+p1z+p2z +• regular in the unit disc A, and for which Re/?(z)>0. In a recent paper [2] it was shown that if p(z)=l+p1z+p2z +• • is regular in A, with p(z)^0 in A, and if a is a real number, then for z e A (1) Re[/>(z) + x(zp'(z)lp(z))] > 0 => Re/7(z) > 0; thatis,/?(z)e^. In this note we replace the differential inequality in (1) by a much more general condition which will still imply that p(z) is a Caratheodory function. DEFINITION 1. Let W=W!+M2/ and v=v1+v2i, and let Y be the set of functions y)(u9 v) satisfying: (a) y)(u, v) is continuous in a domain D of Cx C; (b) ( l , 0 ) e D and Re y(l, 0)>0; (c) Re y)(u2i, vJ^O when (u2i, vx) e D and v^ — 1/2(1 +ui). We denote by O the subset of Y which satisfies (a), (b) and the following condition: , (c') Re y>(u2i, vx)^0 when (u2i, vj e D and i>i^0. EXAMPLES. It is easy to check that each of the following functions are inT. xp^u, V)=U+OLVIU, a real, with D=[C— {0}]xC y,z(u, v)=u +v with D=CxC. y>z(u9 V)=U+OLV, a^O, with D=CxC. n(u, v)=u-vlu 2 with D= [C-{0}] X C. y)5(u, v)=-ln(i-v) with D=Cx{(vl9 u2)h<i}AMS (MOS) subject classifications (1970). Primary 30A04, 30A20, 34A40; Secondary 30A32.