Jakubowski starlike integral operators

LEMMA 2.1. The function f belongs to S(m, M) if and only if there exists a function w regular in U and satisfying w(0) = 0, | w(z) |< 1 for z £ U such that (21) /Xz) = 1 + aw(z) /(z) 1 - bw{z) where a = (M2 - m2 + m)/M and b = (m - \)/M. Next we have the well known Jack's lemma [5].

LEMMA 2.2. / / the function w is regular for \ z |«£ r < 1, w>(0) = 0 and | w(z0) |= ^ l w(z) I, then zow'(zQ) = kw(z0), where k is a real number such that k > 1. Lastly we prove a lemma which plays an important role in establishing one of our main results.

LEMMA 2.3. Let a, /?, m and M be real numbers such that 0 < a < /}, | m — 11< M<m.Ift = (ma + 0 - a)/fi and T = Ma/fi then S(t, T) C S(m, M). PROOF. We need only to consider the case when a < p. In order to establish the lemma it suffices to show that (2.2) m- M<t-T and t+T<m + M.

THEOREM 3.1. Let a, fi, y and 8 be real constants such that 0 < a < ft and y + P = S + a.IffeS(m,M) then the function F defined by (3-D F(z) also belongs to S(m, M), provided y> — [/? + a(m — M — 1)]. In (3.1) all powers are principal ones. PROOF. First of all we show that F £ S(t,T). Let us choose a function w such that F'{z) __ \ + cwjz) F(z) 1 - ew(z) where w(0) = 0 and w is either regular or meromorphic in U; here c = (T2 — t2 + t)/T and e = (t - \)/T. From (3.1) and (3.2) we have (3 3)

COROLLARY 3.1. 7/0 *s a < 1/(1 - m + M), a < j8, and iff e S(w, M), rte function F defined by afao belongs to S(m, M). The above corollary follows by taking y = 1 — j8 and 6 = 1 — a in Theorem 3.1. REMARK. When m = M and m -» oo, a result of Miller ef a/. [9, Theorem 3] follows from this corollary. Another direct but important consequence of Theorem 3.1 is the following result which enables us to obtain the expressions for the functions in S(m, M) of curious forms.

COROLLARY 3.2. Let a and TJ be real constants such that a > 0, TJ > 0. // fES(m,M) then the function F defined by [ v + a + z" [ a/so belongs to S(m, M), provided y + i) S* -a(m — M). l/(o+i) The above result is obtained if, in Theorem 3.1, we take /J = a + TJ and 6 = y + i). For y + Tj = 1, o = 1, TJ = 0,1,2,..., we obtain the sequence j and for y = 0, a = 1, TJ = 0,1,2,..., we obtain the sequence / ( ) ]

Corollary 3.2. Let us choose m = N — p(N — 1) and M = N(l — p), where N > 1 and 0 < p < 1. Then | m - 11< M =s m, a = p/N + (1 - 2p) and b = 1 - 1/W. Now as 7/ -> oo, a -»(1 — 2p) and 6 -» 1. In this case the relation (2.1) reduces to /'(*)_ l + (l-2pHz) z ; j~\ f(z) 1 - w(z) e t/, which is a necessary and sufficient condition for/to be in S*(p). The undermen- tioned corollary follows now from Corollary 3.2.

COROLLARY 3.3. Let a and TJ be real constants such that a > 0, TJ S* 0. // / £ S*(p) then the function F defined by (3.9) also belongs to S*(p) provided y + TJ ^ -ap. REMARKS, (i) A result of Miller et al. [9, Theorem 4] follows by taking p = 0 in the above corollary. (ii) A result of Gupta and Jain [4, Theorem 1] follows by taking TJ = 0 in

Corollary 3.3. However the transform 14/3 = Z + ••• can be studied by Corollary 3.3 and not by the result of Gupta and Jain, since, the technique followed by them fails when a and y are not positive integers. Let - 1 < B < A < 1. If we set m = (1 - AB)/{\ - B2) and M = (A - B)/{\ - B2), then (2.1) becomes ,3,0, f(z) 1 Let us denote by S*[A, B], -1 <£ B <A < 1, the class of functions/ satisfying (3.10). Then, including the limiting case B -» -1 (Corollary 3.3), Theorem 3.1 provides:

COROLLARY 3.4. Let y be any real number such that y > -(1 — A)/(I — B). If f £ S*[A, B], then the function F defined by (3.11) F(z)=l also belongs to S*[A, B].

THEOREM 3.2. Let y be a real number such that y > -(1 - A)/{\ - B). If f E C*[A, B], then the function F defined by (3.11) also belongs to C*[A, B]. PROOF. Since / £ C*[A, B], there exists a function g in S*[A, B] such that Re{zf'(z)/g(z)} > 0, z E U. By Corollary 3.4, for g £ S*[A, B], the function G defined by (3.12) G{z)=*±. also belongs to S*[A, B]. It is easy to obtain from (3.11) and (3.12) that + \ + zF"(z)/F'(z) = ( \ g(z) \ G(z) }[ y +

THEOREM 3.3. / / / G S(m, M) and y > 0, then the function F defined by (3.16) also belongs to S(m, M). PROOF. Let us choose a function w such that (3.17) z = Z F(z) l-bw(z) where w(Q) = 0 and w is either regular or meromorphic in U; here a = (M2 — m2 + m)/M and fc = (m — \)/M. Differentiating (3.15) logarithmically and using (3.17) we get _/'(') _m= (I ~ m) + (a + bm)w(z) + (a + b)zw'(z) f(z) l

THEOREM 4.1. Let a, 0, y, 8 and o be real constants such that a > 0, j8 > a, a>0, a + 8 = 0 + y and y > -(a + /3p). If f £ S*(p) and g e S(m, M), (m,M) EE- {(m,M): \m - 1|< Af < m*}, where m* = min{m,(m- 1) + 0(1 - P)/(2O(Y + a + j8p))}, then the function F defined by (4.D F{Z) also belongs to S*(p). In (4.1) all powers are principal ones. PROOF. Let us choose a function w such that F(z) 1 + w(z) where w(0) = 0 and w is either regular or meromorphic in U.