# Second order non-linear strong differential subordinations.

1 Introduction

Let H = H(U) denote the class of functions analytic in U. For n a positive integer and a [member of] C, let

H[a, n] = {f [member of] H; f (z) = a + [a.sub.n][z.sub.n] + [a.sub.n] + [1.sup.[z.sup.n+1]] + ... , z [member of] U}.

Let A be the class of functions f of the form

f (z) = z + [a.sub.2][z.sup.2] + [a.sub.3][z.sup.3] + ..., z [member of] U,

which are analytic in the unit disk.

In addition, we need the classes of convex, alpha-convex, close-to-convex and starlike (univalent) functions given respectively by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In order to prove our main results we use the following definitions and lemmas.

Definition 1. , ,  Let H(z, [xi]) be analytic in U x [bar.U] and let f (z) analytic and univalent in U. The function H(z, [xi]) is strongly subordinate to f (z), written H(z, [xi]) [??] f(z), if for each [xi] [member of] [bar.U], the function of z, H(z, [xi]) is subordinate to f (z).

Remark 1. (i) Since f (z) is analytic and univalent, Definition 1 is equivalent to:

H(0, [xi]) = f (0) and H(U x U) ? f (U). (ii) If H(z, x) = H(z) then the strong subordination becomes the usual subordination.

Definition 2. , [5, p.21] We denote by Q the set of functions q that are analytic and injective in [bar.U] \ E(q), where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and are such that q'([xi]) [not equal to] 0 for [xi] [member of] U \ E(q).

The subclass of Q for which f (0) = a is denoted by Q(a).

Lemma A. [5, Lemma 2.2.d, p.24] Let q [member of] Q(a), with q(0) = a and p(z) = a + [a.sub.n][z.sup.n] + [a.sub.n] + [1.sup.[z.sup.n+1]] + ... be analytic in U, with p(z) [??] a and n [greater than or equal to] 1. If p is not subordinate to q, then there exist points [z.sub.0] = [r.sub.0][e.sup.i[[theta].sub.0]] [member of] U and [[xi].sub.0] [member of] [partial derivative]U \ E(q), and an m [greater than or equal to] n [greater than or equal to] 1 for which p([U.sub.[r.sub.0]]) [subset] q(U),

(i) p([z.sub.0]) = q([[xi].sub.0)]

(ii) [z.sub.0]p'([z.sub.0]) = m[[xi].sub.0]q'([[xi].sub.0)], and

(iii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Definition 3. [7, Definition 4] Let [Omega] be a set in C, q [member of] Q and n be a positive integer. The class of admissible functions yn[W, q] consists of those functions [psi]: [C.sup.3] x U x [bar.U] [right arrow] C that satisfy the admissibility condition:

(A) [psi](r, s, t; z, [xi]) [member of] W

whenever r = q([zeta]), s = m[zeta]q'([zeta]),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and m [greater than or equal to] n.

Remark 2. For the function q(z) = Mz, M > 0, z [member of] U, the condition of admissibility (A) becomes

(A') [psi]([Me.sup.i[theta]], [Ke.sup.i[theta]], L; z, [xi]) [member of] W

whenever [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [theta] [member of] R.

For the function q(z) = 1 + z / 1 - z, z [member of] U, the condition of admissibility (A) becomes

(A") [psi]([rho]i, [sigma], [mu] + vi; z, [xi]) [member of] [Omega]

whenever [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

2 Main results Definition 4. A strong differential subordination of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a function of z, analytic for all [xi] [member of] U and function h is analytic and univalent in U, is called second order non-linear strong differential subordination.

Remark 3. If D(z, [xi]) = 0 then we obtain a second order linear strong differential subordination studied in .

Remark 4. For A(z, [xi]) = D(z, [xi]) = 0 the second order non-linear strong differential subordination reduces to the first order linear differential subordination studied in .

Theorem 1. Let A, B, C, D, E : U x [bar.U] [right arrow] C with

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] a function of z analytic for all [xi] [member of] U.

If p [member of] H[0, n] and the second order non-linear strong differential subordination

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] holds, then

p(z) [??] Mz, z [member of] U, M > 0.

Proof. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], let

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then (2) becomes

(4) [psi](r, s, t; z, [xi]) [??][??] Mz, z [member of] U, [xi] [member of] [bar.U].

If we let h(z) = Mz, z [member of] U, M > 0 then h(U) = U(0,M) and (4) is equivalent to

(5) [psi](r, s, t; z, [xi]) [member of] U(0,M), z [member of] U, [xi] [member of] [bar.U].

Suppose that p is not subordinate to function h. Then, by Lemma A, we have that there exist [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [[zeta].sub.0] [member of] [partial derivative]U such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where z [member of] U, [[theta].sub.0] [member of] R.

By replacing r with p([z.sub.0]), s with [z.sub.0]p'([z.sub.0]), t with [z.sup.2.sub.0]p"([z.sup.0]) in (3) and using the conditions given by (1) we obtain

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since (6) contradicts (5), the assumption made is false and hence, p(z) [??] Mz, z [member of] U, M > 0.

Example 1. Let

A(z, [xi]) = 2, B(z, [xi]) = z + [xi] + 3. 2i, C(z, [xi]) = 2z + [xi] + 5 - i, D(z, [xi]) = z + [xi] + 2, E(z, [xi]) = 0, n = 1, M = 1/4, z [member of] U, [xi] [member of] [bar.U].

Since z [member of] U, [xi] [member of] [bar.U], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

From Theorem 1, we obtain:

If

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is a function of z, analytic for all [xi] [member of] [bar.U] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then

p(z) [??] z, z [member of] U.

Theorem 2. Let A, B, C, D, E : U x [bar.U] [right arrow] C with

(7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be an analytic function of z for all [xi] [member of] [bar.U].

If p [member of] H[1, n] and the following second order strong differential subordination holds

(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then

p(z) [??] 1 + z / 1 - z, z [member of] U.

Proof. Let [psi]: [C.sup.3] x U x [bar.U] [right arrow] C and for r = p(z), s = zp'(z), t = [z.sup.2]p"(z) we have

(9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then (8) becomes

(10) [psi](r, s, t; z, [xi]) 1 + z/1 - z, z [member of] U, [xi] [member of] [bar.U].

If we let q(z) = 1 + z/1 - z, z [member of] U then h(U) = {w [member of] C; Re w > 0}, the strong differential subordination (10) implies

(11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and (11) implies

(12) Re [psi](r, s, t; z, [xi]) > 0, z [member of] U, [xi] [member of] [bar.U].

Suppose that p is not subordinate to the function q(z) = 1 + z/1 - z, z [member of] U. Then, by Lemma A, we have that there exist points [z.sub.0] [member of] U, [z.sub.0] = [r.sub.0][e.sup.i[theta].sub.0]], [[theta].sub.0] [member of] R and [[zeta].sub.0] [member of] [partial derivative]U such that p([z.sub.0]) = q([[zeta].sub.0]) = [rho]i, [rho] [member of] R, [z.sub.0]p'([z.sub.0]) = m[zeta]0q'([[zeta].sub.0]) = [sigma], [sigma] [member of] R, [z.sup.2.sub.0]p"([z.sub.0]) = [[zeta].sup.2.sub.0]q"([[zeta].sub.0]) = [mu] + iv, [mu], v [member of] R with [sigma] [less than or equal to] - n/2 (1 + [[rho].sup.2]) and [sigma] + [mu] [less than or equal to] 0, m [greater than or equal to] n [greater than or equal to] 1.

By replacing r = p([z.sub.0]), s = [z.sub.0]p'([z.sub.0]), t = [z.sup.2.sub.0]p"([z.sub.0]) in (9) and using the conditions given by (7), we obtain

(13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

p(z) [??] 1 + z/1 - z, z [member of] U.

Remark 5. Theorem 2 can be rewritten as follows:

Corollary 1. Let A, B, C, D, E : U x [bar.U] [right arrow] C, n [member of] N,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

a function of z, analytic for all [xi] [member of] [bar.U] with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If p [member of] H[1, n] and satisfies the inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then

Re p(z) > 0, z [member of] U.

Remark 6. Note that the result contained in Theorem 2 can be applied to obtain sufficient conditions for univalence on the unit disc, such as starlikeness, convexity, alpha-convexity, close-to-convexity. Indeed, it suffices to consider

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and p(z) = f '(z), z [member of] U respectively.

Example 2. A(z, [xi]) = 2, B(z, [xi]) = z + [xi] + 6 - 2i, C(z, [xi]) = -z - [xi] + 1 - 10i, D(z, [xi]) = 2z + [xi] + 3+ 10i, E(z, [xi]) = z - 3, n = 2.

Since z [member of] U, [xi] [member of] U, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From Theorem 2, we obtain:

If

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is a function of z, analytic for all [xi] [member of] [bar.U] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Received by the editors January 2008. Communicated by Y. Felix.

References

 Jose A. Antonino and Salvador Romaguera, Strong differential subordination to Briot-Bouquet differential equations, Journal of Differential Equations 114(1994), 101-105.

 Jose A.Antonino, Strong differential subordination and applications to univalency conditions, J. Korean Math. Soc., 43(2006), no.2, 311-322.

 Jose A. Antonino, Strong differential subordination to a class of first order differential equations (to appear).

 S.S. Miller and P.T. Mocanu, Differential subordinations and univalent functions, Michigan Math. J., 28(1981), no. 2, 157-172.

 S.S. Miller and P.T. Mocanu, Differential subordinations. Theory and applications, Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 2000.

 Georgia Irina Oros, Sufficient conditions for univalence obtained by using second order linear strong differential subordinations, accepted for publication in Turkish Journal of Mathematics.

 Georgia Irina Oros and Gheorghe Oros, Strong differential subordination, to appear in Turkish Journal of Mathematics, 32(2008).

 Georgia Irina Oros and Gheorghe Oros, First order strong linear differential subordinations, General Mathematics vol. 15, No. 2-3 (2007), p. 98-107.

Georgia Irina Oros, Gheorghe Oros

Department of Mathematics