# 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. [1], [2], [3] 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. [4], [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 [6].

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 [8].

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

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

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

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

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

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

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

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

[8] 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

Str. Universitatii, No. 1