Some applications of differential subordination

LEMMA 1. Let $3yeC3he H(E) be convex univalent in E with h(0) = 1 and Re(B?z(3)+Y) > 0 , z e E and let q e H(E) with q(0) = 1 and q{z) -e h(s) , z e E . If p(z) = 1 + p z + ... is analytic in E , then % 2 ! y < Mz) " p ( 2 ) <Mz) • The proof is essentially the same as that of Theorem A. However we give the details below for the sake of completeness.$

THEOREM 1. If f e S [h) , then f e 5fl(/2) /zoZds /or a > 1 (X */) (2) provided ^ 0 for z e E . 3 ziK */)'((2)

THEOREM 2. Suppose f e SQ(^) ; then F e Sa(h) provided (K * F) (s) — ^ — f 0 /or 3 e ff . 2

Theorem 1 of R.J. Libera (1965). 1 + A p REMARK 4 . I f a = 1 and h{z) = , „ , - 1 < A < B < 1 , we 1 + uZ deduce Lemma 2 of R.M. Goel and B.S. Mehrok [4]. DEFINITION 3. Let C {h) (h as above) denote the class of functions z(K * f)'{z) f £ A such that * <t>) (2) "* ^ ( Z ) f o r s o m e * e 5a(?l) " REMARK 5. If a = 1 and 7z(2) = j" ~ ^ , then C lh) = C the class of close-to-convex functions introduced by W. Kaplan [5].

THEOREM 3. If f e C (h) 3 then f e Ca<Ji) holds for a > 1 (K * *) (2) provided f 0 for z e E . z z(K * f) '(z)

THEOREM 4. Suppose f e C {h) with respect to the function 4> € Sa(fc) • Define ifi ij/ ijj(3) = (<(> * ^ ) (z) . T?zen F e C Q ( ^ ) with (if * i>) (3) respect to ty , provided

Lemma 1, p(z) < h(z) and hence F e ca^ • 1—3 REMARK 6. For a = 1, r = 1 and h(z) = , this theorem x ~ z reduces to Theorem 3 in R.J. Libera [6]. DEFINITION 4. Let C^(.h) , a > 0 denote the class of functions / e A such that */)'C3) z(K *f)'(z) + (1°° (X * (») (3) for some $ e S (h) satisfying ^ 0 for z e E .$

THEOREM 5. If f e c£(fc) , t?zen f e ti^ih) = CaCft) , for a > 1 Z / ) ( 3 )

THEOREM 6. For a > 6 > 0 <^(/z) <= C^(h) .

Theorem 5 reduces to Theorem 1 of V.A. Zmorovich and V.A. Pokhilevich [9]. DEFINITION 5. Let Sa(h) , a > 0 denote the class of functions f e A such that f)'(z) z(Ka*f)'(z) + {1) {Ka+l*fUz) (K *f)(z) with / 0 and / 0 for z e E . %

THEOREM 7. If f e S*(h) then f c S°(h) = S (h) , for a > 1 . https://doi.org/10.1017/S0004972700002410 Published online by Cambridge University Press

THEOREM 8. For a > B > 0 , S*(h) c 53(^) .

Theorem 7, then we get Theorem 1 of Al-flmiri [ J ] . REMARK 1 0 . I f a = 1 and 7z(3) = 2 then Theorem 7 reduces to 1 + 3 the well-known result that all a-convex functions are starlike by S.S. Miller, P.T. Mocanu and M.O. Reade [7]. References [/] S. Hassoon, Al-Amiri, "Certain analogy of the a-convex functions". Rev. Roum. Math. Pures et Appl. 23 (1978), 1449-1454. [2] P.N. Chichra, "New subclasses of the class of close-to-convex functions", Proa. Amer. Math. Soo. 62 (1977), 37-43. [3]