# On special strong differential subordinations using a multiplier transformation and Ruscheweyh derivative.

1. INTRODUCTION

Denote by U the unit disc of the complex plane U = {z G C: \z\ < 1}, [bar.U] = {z [member of] C: [absolute value of z] [less than or equal to] 1} the closed unit disc of the complex plane and H(U x [bar.U]) the class of analytic functions in U x [bar.U].

Let [A.sup.*.sub.n[zeta]] = {f [member of] H(U x [bar.U]), f(z, [zeta]) = z + [a.sub.n+1]([zeta])[z.sup.n+1] + ***, z [member of] U, [zeta] [member of] [bar.U]}, with [A.sup.*.sub.1[zeta]] = [A.sup.*.sub.[zeta]], where [a.sub.k]([zeta]) are holomorphic functions in [bar.U] for k [greater than or equal to] 2, and

[H.sup.*][a, n, [zeta]] = (f [member of] H(U x [bar.U]), f(z, [zeta]) = a + [a.sub.n]([zeta])[z.sup.n] + [a.sub.n+1](C)[z.sup.n+1] + ***, z [member of] U, [zeta] [member of] [bar.U]},

for a [member of] C, n [member of] N, [a.sub.k]([zeta]) are holomorphic functions in U for k > n.

Generalizing the notion of differential subordinations, J.A. Antonino and S. Romaguera have introduced in [10] the notion of strong differential subordinations, which was developed by G. I. Oros and Gh. Oros in [11], [12].

Definition 1.1 ([12]). Let f(z, C), H(z, C) analytic in U x [bar.U]. The function f(z, [zeta]) is said to be strongly subordinate to H(z, [zeta]) if there exists a function w analytic in U, with w (0) = 0 and [absolute value of w(z)] < 1 such that f(z, [zeta]) = H(w(z), [zeta]) for all [zeta] [member of] [bar.U]. In such a case we write f(z, [zeta]) [??][??] (z, C), z [member of] U, [zeta] [member of] [bar.U].

Remark 1.1 ([12]). i) Since f(z, [zeta]) is analytic in U x [bar.U], for all [zeta] [member of] [bar.U], and univalent in U, for all [zeta] [member of] [bar.U], Definition 1.1 is equivalent to f(0, [zeta]) = H(0, [zeta]), for all [zeta] [member of] [bar.U], and f(U x [bar.U]) [subset] H(U x [bar.U]) .

ii) If H (z, [zeta]) [equivalent to] H(z) and f(z, [zeta]) [equivalent to] f(z), the strong subordination becomes the usual notion of subordination.

We need the following lemmas to study the strong differential subordinations.

Lemma 1.1 ([4]). Let h(z, [zeta]) he a convex function with h(0, [zeta]) = a for every [zeta] [member of] [bar.U] and let [gamma] [member of] [C.sup.*] be a complex number with Re [gamma] [greater than or equal to] 0.

If p [member of] [H.sup.*][a, n, [zeta]] and p(z, [zeta]) + [1/[gamma]]z[p'.sub.z](z, [zeta]), [??][??] h{z, [zeta]), then

p(z, [zeta]) [??][??] g(z, [zeta]) [??][??] h (z, [zeta]),

where g(z, [zeta]) = [gamma]/[nz.sup.[gamma]/n] [[integral].sup.z.ub.0]h(t, [zeta])[t.sup.[[gamma]/n-1]] dt is convex and it is the best dominant.

Lemma 1.2 ([4]). Let g (z, C) be a convex function in U x [bar.U], for all [zeta] [member of] [bar.U], and let

h(z, [zeta]) = g(z, [zeta]) + [n[alpha]zg'.sub.z](z, [zeta]), z [member of] U, [zeta] [member of] [bar.U],

where [alpha] > 0 and n is a positive integer.

If p(z, [zeta]) = g(0, [zeta]) + [p.sub.n] ([zeta]) [z.sup.n] + [P.sup.n+1] ([zeta]) [z.sup.n+1] + ***, z [member of] U, [zeta] [member of] U, is holomorphic in U x [bar.U] and p(z, [zeta]) + [alpha]z[p'.sub.z](z, [zeta]) [??][??] h{z, [zeta]), z [member of] U, [zeta] [member of] [bar.U], then

p(z, [zeta]) [??][??] g(z, [zeta])

and this result is sharp.

We also extend the Ruscheweyh derivative [13] and the multiplier transformation [2] to the new class of analytic functions [A.sup.*.sub.n[zeta]] introduced in [11].

Definition 1.2 ([4]). For f [member of] [A.sup.*.sub.n[zeta]] and n, m [member of] N, the operator [R.sup.m] is defined by [R.sup.m]: [A.sup.*.sub.n[zeta]] [right arrow] [A.sup.*.sub.n[zeta]]

[R.sup.0]f (z, [zeta]) = f(z, [zeta]), [R.sup.1]f (z, [zeta]) = z[f'.sub.z](z, [zeta])

(m + 1) [R.sup.m+1]f(z, [zeta]) = z[([R.sup.m]f(z, [zeta]))'.sub.z] + m[R.sup.m]f (z, [zeta]), z [member of] U, [zeta] [member of] [bar.U].

Remark 1.2 ([4]). If f [member of] [A.sup.*.sub.n[zeta]], f(z, [zeta]) = z + [[infinity].summation over (j=n+1)] [a.sub.j] ([zeta]) [z.sup.j], then

[R.sup.m]f(z, [zeta]) = z + [[infinity].summation over (j=n+1)] [C.sup.m.sub.m+j-1]aj ([zeta]) [z.sup.j], z [member of] U, [zeta] [member of] [bar.U].

Definition 1.3 ([6]). For n [member of] N, m [member of] N [union]{0}, [lambda], l [greater than or equal to] 0, f [member of] [A.sup.*.sub.n[zeta]],

f(z, [zeta]) = z + [[infinity].summation over (j=n+1)][a.sub.j] (Z) [z.sup.j],

the operator I (m, [lambda], l) f (z, [zeta]) is defined by the following infinite series

I(m, [lambda], l)f(z, [zeta]) = z + [[infinity].j=n+1] [(1 + [lambda](j - 1) + l/l + 1).sup.m] [a.sub.j]([zeta])[z.sup.j], z [member of] U, [zeta] [member of] [bar.U].

Remark 1.3 ([6]). It follows from the above definition that

(l + 1)I(m + 1, [lambda], l)f(z, [zeta]) = [l + 1 - [lambda]]I(m, [lambda], l)f(z, [zeta]) + [lambda]z(I(m, [lambda], l)f[(z, [zeta]))'.sub.z],

whenever z [member of] U and [zeta] [member of] U.

We extend the differential operator studied in [1], [3] to the new class of analytic functions [A.sup.*.sub.n[zeta]].

Definition 1.4. Let [alpha], [lambda], l [greater than or equal to] 0, m [member of] N. Denote by [RI.sup.[alpha].sub.m,[lambda],l] the operator given by [RI.sup.[alpha].sub.m,[lambda],l]: [A.sup.*.sub.[zeta]] [right arrow] [A.sup.*.sub.[zeta]],

[RI.sup.[alpha].sub.m,[lambda],l]f(z, [zeta]) = (1 - [alpha])[R.sup.m]f(z, [zeta]) + [alpha]I(m, [lambda], l)f(z, [zeta]), z [member of] U, [zeta] [member of] [bar.U].

Remark 1.4. If f [member of] [A.sup.*.sub.[zeta]], f(z, [zeta]) = z + [[infinity].summation over (j=2)][a.sub.j] ([zeta]) [z.sup.j], then

[RI.sup.[alpha].sub.m,[lambda],l]f(z, [zeta]) = z + [[infinity].summation over (j=2)]{[alpha][(1 + [lambda](j - 1) + l/l + 1).sup.m] + (1 - [alpha])[C.sup.m.sub.m+j-1]}[a.sub.j] (zeta) [z.sup.j],

z [member of]U, [zeta] [member of] [bar.U].

Remark 1.5. For [alpha] = 0, [RI.sup.0.sub.m,[lambda],l]f(z, [zeta]) = [R.sup.m]f(z, [zeta]), where z [member of] U, [zeta] [member of] [bar.U], and for [alpha] = 1, [RI.sup.0.sub.m,[lambda],l]f(z, [zeta]) = I (m, [lambda], l)f(z, [zeta]), where z [member of] U, [zeta] [member of] [bar.U], which was studied in [6], [7].

For l = 0, we obtain [RI.sup.[alpha].sub.m,[lambda],l]f(z, [zeta]) = [RI.sup.m.sub.1,[alpha]]f(z, [zeta]) which was studied in [8], [9] and for l = 0 and [lambda] = 1, we obtain [RI.sup.[alpha].sub.m,[lambda],0]f(z, [zeta]) = [L.sup.m.sub.[alpha]]f(z, [zeta]) which was studied in [4], [5].

For m = 0, [RI.sup.[alpha].sub.0,[lambda],l]f(z, [zeta]) = (1 - [alpha]) [R.sup.0]f(z, [zeta]) + [alpha]l(0, [lambda], l)f(z, [zeta]) = f(z, [zeta]) = [R.sup.0]f(z, [zeta]) = I (0, [lambda], l)f(z, [zeta]), where z [member of] U, [zeta] [member of] [bar.U].

2. MAIN RESULTS

Definition 2.1. Let [delta] [member of] [0,1), [alpha], [lambda], l [greater than or equal to] 0and m [member of] N. A function f (z, [zeta]) [member of] [A.sup.*.sub.[zeta]] is said to be in the class RJ([delta], [lambda], l, [alpha], [zeta]) if it satisfies the inequality

Re[([RI.sup.[alpha].sub.m,[lambda],l](z, [zeta]))'.sub.z] > [delta], z [member of] U, [zeta] [member of] [bar.U]. (2.1)

Theorem 2.1. The set [RJ.sub.m] ([delta], [lambda], l, [alpha], [zeta]) is convex.

Proof. Let the functions [f.sub.j](z, [zeta]) = z + [[infinity].summation over (j=2)][a.sub.jk] ([zeta]) [z.sup.j], k = 1, 2, z [member of] U, [zeta] [member of] [bar.U], be in the class

[RJ.sub.m] ([delta], [lambda], l, [alpha], [zeta]). It is sufficient to show that the function

h(z, [zeta]) = [[eta].sub.1][f.sub.1] (z, [zeta]) + [[eta].sub.2][f.sub.2] (z, [zeta])

is in the class [RJ.sub.m] ([delta], [lambda], l, [alpha], [zeta]), with [[eta].sub.1] and [[eta].sub.2] nonnegative such that [[eta].sub.1] + [[eta].sub.2] = 1.

Since h (z, [zeta]) = z + [[infinity].summation over (j=2)]([[eta].sub.1][a.sub.j1] ([zeta]) + [[eta].sub.2][a.sub.j2] ([zeta]))[z.sup.j], z [member of] U, [zeta] [member of] U, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.2)

Differentiating (2.2) with respect to z we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.3)

Taking into account that [f.sub.1], [f.sub.2] [member of] [RJ.sub.m] ([delta], [lambda], l, [alpha], [zeta]) we deduce

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.4)

Using (2.4) we get from (2.3)

Re [([RI.sup.[alpha].sub.m,[lambda],l]h(z, [zeta])'.sub.z] > 1 + [[eta].sub.1]([delta] - 1) + [[eta].sub.2]([delta] - 1) = [delta], z [member of] U, [zeta] [member of] [bar.U],

which is equivalent that [RJ.sub.m]([delta], [lambda], l, [alpha], [zeta])is convex.

Theorem 2.2. Let g(z, [zeta]) be a convex function such that g(0, [zeta]) = 1 and let h (z, [zeta]) = g(z, [zeta])+ [1/c + 2][zg'.sub.z](z, [zeta]), where z [member of] U, [zeta] [member of] [bar.U], c > 0.

If [alpha], [lambda], l [greater than or equal to] 0, m [member of] N, f [member of] [RJ.sub.m] ([delta], [lambda], l, [alpha], [zeta]) and

F(z, [zeta] = [I.sub.c](f)(z, [zeta]) = [c + 2/[z.sup.c+1]] [[integral].sup.z.sub.0] [t.sup.c]f(t, [zeta])dt, z [member of] U, [zeta] [member of] [bar.U],

then

([RI.sup.[alpha].sub.m,[lambda],l]F(z, [zeta]))'z [??][??] h(z, [zeta]), z [member of] U, [zeta] [member of] [bar.U], (2.5)

implies [([RI.sup.[alpha].sub.m,[lambda],l]F(z, [zeta]))'.sub.z] [??][??] g(z, [zeta]), z [member of] U, [zeta] [member of] [bar.U], and this result is sharp.

Proof. We obtain that

[z.sup.c+1]F(z, [zeta]) = (c + 2) [[integral].sup.z.sub.0] [t.sup.c]f(t, [zeta])dt. (2.6)

Differentiating (2.6), with respect to z, we have

(c + 1) F(z, [zeta]) + z[F'.sub.z](z, [zeta]) = (c + 2)f(z, [zeta])

and

(c + 1)[RI.sup.[alpha].sub.m,[lambda],l]F(z, [zeta]) + z[([RI.sup.[alpha].sub.m,[lambda],l]F(z, [zeta]))'.sub.z] = (c + 2) [RI.sup.[alpha].sub.m,[lambda],l](z, [zeta]), z [member of] U, [zeta] [member of] [bar.U]]. (2.7)

Differentiating (2.7) with respect to z we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.8)

Using (2.8), the strong differential subordination (2.5) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.9)

Denote

p(z, [zeta]) = [([RI.sup.[alpha].sub.m,[lambda],l]F(z, [zeta])'.sub.z], z [member of] U, [zeta] [member of] [bar.U]. (2.10)

Replacing (2.10) in (2.9) we obtain

p(z, [zeta]) + [1/c + 2][zp'.sub.z] (z, [zeta]) [??][??] g(z, [zeta]) + [1/c + 2][zg'.sub.z] (z, [zeta]) z [member of] U, [zeta] [member of] [bar.U].

Using Lemma 1.2 we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and this result is sharp.

Theorem 2.3. Let h(z, [zeta]) = [[zeta] + 2[delta] - [zeta])z/1 + z], z [member of] U, [zeta] [member of] [bar.U], [delta] [member of] [0, 1) and c > 0. If [alpha], [lambda], l [greater than or equal to] 0, m [member of] N and [I.sub.c] is given by Theorem 2.2,then

[I.sub.c] [[RJ.sub.m]([delta], [lambda], l, [alpha], [zeta])] [subset] [RJ.sub.m] ([delta]*, [lambda], l, [alpha], [zeta]), (2.11)

where [delta]* = 2[delta] - [zeta] + [2(c + 2)([zeta] - [delta])/n][beta](c + 1) and [beta](x) = [[integral].sup.1.sub.0] [[t.sup.x]/t + 1]dt.

Proof. The function h is convex and using the same steps as in the proof of Theorem 2.2 we get from the hypothesis of Theorem 2.3 that

p(z, [zeta]) + [1/c + 2][zp.sub.'z](z, [zeta]) [??][??] h(z, [zeta]),

where p(z, [zeta]) is defined in (2.10).

Using Lemma 1.1 we deduce that p(z, [zeta]) [??][??] g(z, [zeta]) [??][??] h(z, [zeta]),that is

[([RI.sup.[alpha].sub.m,[lambda],l]F(z, [zeta]))'.sub.z] [??][??] g(z, [zeta]) [??][??] h(z, [zeta]),

where

g(z, [zeta]) = [c + 2/[z.sup.c+2]] [[integral].sup.z.sub.0] [t.sup.c+1][[zeta] + (2[delta] - [zeta])t/1 + t]dt = (2[delta] - [zeta]) + [2(c + 2)([zeta] - [delta])/[z.sup.c+2]] [[integral].sup.z.sub.0] [[t.sup.c+1]/1 + t]dt.

Since g is convex and g(U x [bar.U]) is symmetric with respect to the real axis, we deduce

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.12)

From (2.12) we deduce inclusion (2.11).

Theorem 2.4. Let g(z, [zeta]) be a convex function such that g(0, [zeta]) = 1and let h be the function h(z, [zeta]) = g(z, [zeta]) + [zg'.sub.z](z, [zeta]), z [member of] U, [zeta] [member of] [bar.U].

If [alpha], [lambda], l [greater than or equal to] 0, m [member of] N, f [member of] [A.sup.*.sub.[zeta]] and satisfies the strong differential subordination

[([RI.sup.[alpha].sub.m,[lambda],l]f(z, [zeta]))'.sub.z] [??][??] h(z, [zeta]), z [member of] U, [zeta] [member of] [bar.U], (2.13)

then [[RI.sup.[alpha].sub.m,[lambda],l]f(z, [zeta])/z] [??][??] g(z, [zeta]), z [member of] U, [zeta] [member of][bar.U] and this result is sharp.

Proof. By using the properties of operator [RI.sup.[alpha].sub.m,[lambda],l], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Consider

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let [RI.sup.[alpha].sub.m,[lambda],l]f(z, C) = zp(z, [zeta]), z [member of] U, [zeta] [member of] [bar.U]. Differentiating with respect to z we obtain [([RI.sup.[alpha].sub.m,[lambda],l]f(z, [zeta]))'.sub.z] = p(z, [zeta]) + [zp'.sub.z](z, [zeta]) + z [member of] U, [zeta] [member of] [bar.U].

Then (2.13) becomes

p(z, [zeta]) + [zp'.sub.z](z, [zeta]) [??][??] h(z, [zeta]) = g(z, [zeta]) + [zg'.sub.z](z, [zeta]), z [member of] U, [zeta] [member of] [bar.U].

By using Lemma 1.2, we have

P(z, [zeta]) [??][??] g(z, [zeta]), z [member of] U, [zeta] [member of] [bar.U], i.e. [[RI.sup.[alpha].sub.m,[lambda],l]f(z, [zeta])/z] [??][??] g(z, [zeta]),

whenever z [member of] U and [zeta] [member of] [bar.U].

Theorem 2.5. Let h (z, [zeta]) be a convex function such that h(0, [zeta]) = 1. If [alpha], [lambda], l [greater than or equal to] 0, m [member of] N, f [member of] [A.sup.*.sub.[zeta]] and satisfies the strong differential subordination

[([RI.sup.[alpha].sub.m,[lambda],l]f(z, [zeta]))'.sub.z] [??][??] h(z, [zeta]), z [member of] U, [zeta] [member of] [bar.U], (2.14)

then

[[RI.sup.[alpha].sub.m,[lambda],l] f(z, [zeta])/z] [??][??] g(z, [zeta]) [??][??] h(z, [zeta]), z [member of] U, [zeta] [member of] [bar.U],

where g(z, [zeta]) = 1/z h(t, [zeta])dt is convex and it is the best dominant.

Proof. Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Differentiating with respect to z, we obtain

[[RI.sup.[alpha].sub.m,[lambda],l]f(z, [zeta])'.sub.z] = p(z, [zeta]) + [zp'.sub.z](z, [zeta]), z [member of] U, [zeta] [member of] [bar.U],

and (2.14) becomes p(z, [zeta]) + [zp'.sub.z](z, [zeta]) [??][??] h(z, [zeta]), z [member of] U, [zeta] [member of] U.

Since p(z, [zeta]) [member of] [H.sup.*][1, 1, [zeta]], using Lemma 1.1 for n = 1 and [gamma] = 1, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and g(z, [zeta]) is convex and it is the best dominant.

Corollary 2.1. Let h(z, [zeta]) = [[zeta] + (2[beta] - [zeta])z/1 + z] convex function in U x [bar.U], 0 [less than or equal to] [beta] < 1. If [alpha] [greater than or equal to] 0, m [member of] N, f [member of] [A.sup.*.sub.[zeta]] and verifies the strong differential subordination

[([RI.sup.[alpha].sub.m,[lambda],l]f(z, [zeta]))'.sub.z] [??][??] h(z, [zeta]), z [member of] U, [zeta] [member of] [bar.U], (2.15)

then [[RI.sup.[alpha].sub.m,[lambda],l]f(z, [zeta])/z] [??][??] g(z, [zeta]) [??][??] h(z, [zeta]), z [member of] U, [zeta] [member of] [bar.U], where g is given by g(z, [zeta]) = 2[beta] - [zeta] + [2([zeta] - [beta])/z]ln (1 + z), z [member of] U, [zeta] [member of] [bar.U]. The function q is convex and it is the best dominant.

Proof. Following the same steps as in the proof of Theorem 2.5 and considering p(z, [zeta])= [RI.sup.[alpha].sub.m,[lambda],l]f(z, [zeta])/z, the strong differential subordination (2.15) becomes

P(z, [zeta]) + [zp'.sub.z](z, [zeta]) [??][??] h(z, [zeta]) = [[zeta] + (2[beta] - [zeta])z/1 + z], z [member of] U, [zeta] [member of] [bar.U].

By using Lemma 1.1 for n = 1 and [gamma] = 1, we have p(z, [zeta]) [??][??] g(z, [zeta]) [??][??] h(z, [zeta]), z [member of] U, [zeta] [member of] [bar.U], i.e.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and this completes the proof.

Theorem 2.6. Let g(z, [zeta]) be a convex function such that g(0, [zeta]) = 1 and let h be the function h (z, [zeta]) = g(z, [zeta]) + [zg'.sub.z](z, [zeta]), z [member of] U, [zeta] [member of] [bar.U].

If [alpha], [lambda], l [greater than or equal to] 0, m [member of] N, f [member of] [A.sup.*.sub.[zeta]] and the strong differential subordination

[(z[RI.sup.[alpha].sub.m+1,[lambda],l]f(z, [zeta])/[RI.sup.[alpha].sub.m,[lambda],l]f(z, [zeta]))'.sub.z] [??][??] h(z, [zeta]), z [member of] U, [zeta] [member of] [bar.U], (2.16)

holds, then [RI.sup.[alpha].sub.m+1,[lambda],l]f(z, [zeta])/[RI.sup.[alpha].sub.m,[lambda],l]f(z, [zeta]) [??][??] g(z, [zeta]), z [member of] U, [zeta] [member of] [bar.U], and this result is sharp.

Proof. For f [member of] [A.sup.*.sub.[zeta]] f(z, [zeta]) = z + [[infijnity].summation over (j=2)][a.sub.j] ([zeta]) [z.sup.j] we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Consider

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and we obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Relation (2.16) becomes

p(z, [zeta] + [zp'.sub.z](z, [zeta]) [??][??] h(z, [zeta]) = g(z, [zeta]) + [zg'.sub.z](z, [zeta]), z [member of] U, [zeta] [member of] [bar.U].

By using Lemma 1.2, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The proof is complete.

Theorem 2.7. Let g(z, [zeta]) be a convex function such that g(0, [zeta]) = 1 and let h be the function h(z, [zeta]) = g(z, [zeta]) + [zg'.sub.z](z, [zeta]), z [member of] U, [zeta] [member of] U.

If [alpha], [lambda], l [greater than or equal to] 0, m [member of] N, f [member of] [A.sup.*.sub.[zeta]] and the strong differential subordination

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.17)

holds, then

[[RI.sup.[alpha]m,[lambda],l]f(z, [zeta]) [??][??] g(z, [zeta]), z [member of] U, [zeta] [member of] [bar.U].

This result is sharp.

Proof. Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.18)

By using the properties of operators [RI.sup.[alpha].sub.m,[lambda],l], [R.sup.m] and I(m, [lambda], l), after a short calculation, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Using the notation in (2.18), the differential subordination becomes

p(z, [zeta]) + [zp'.sub.z](z, [zeta]) [??][??] h(z, [zeta]) = g(z, [zeta])+ [zg'.sub.z](z, [zeta])

By using Lemma 1.2, we have

p(z, [zeta]) [??][??] g(z, [zeta]), z [member of] U, [zeta] [member of] [bar.U], i.e. [([RI.sup.[alpha].sub.m,[lambda],l]f(z, [zeta])'.sub.z] [??][??] g(z, [zeta]), z [member of] U, [zeta] [member of] [bar.U], and this result is sharp.

Theorem 2.8. Let h(z, [zeta]) be a convex function such that h (0, [zeta]) = 1. If [alpha], [lambda], l [greater than or equal to] 0, m [member of] N, f [member of] [A.sup.*.sub.[zeta]] and satisfies the strong differential subordination

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.19)

then

[([RI.sup.[alpha].sub.m,[lambda],l]f(z, [zeta])'.sub.z] [??][??] g(z, [zeta]) [??][??] h (z, [zeta]), z [member of] U, [zeta] [member of] [bar.U],

where g(z, [zeta]) = [1/z] [[integral].sup.z.sub.0] h(t, [zeta])dt is convex and it is the best dominant.

Proof. Using the properties of operator [RI.sup.[alpha].sub.m,[lambda],l] and considering

p(z, [zeta]) = [([RI.sup.[alpha].sub.m,[lambda],l]f(z, [zeta])'.sub.z],

we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then (2.19) becomes

p(z, [zeta]) + [zp'.sub.z](z, [zeta]) [??][??] h(z, [zeta]), z [member of] U, [zeta] [member of] [bar.U].

By using Lemma 1.1, for n = 1 and [gamma] =1, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and g(z, [zeta]) is convex and it is the best dominant.

Corollary 2.2. Let h(z, [zeta] = [[zeta] + (2[beta] - [zeta])/1 + z] a convex function in U x [bar.U], 0 [less than or equal to] [beta] < 1. If [alpha] [greater than or equal to] 0, m [member of] N, f [member of] [A.sup.*.sub.[zeta]] and verifies the strong differential subordination

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.20)

then

[([RI.sup.[alpha].sub.m,[lambda],l]f(z, [zeta])'.sub.z] [??][??] g(z, [zeta]) [??][??] h(z, [zeta]), z [member of] U, [zeta] [member of] [bar.U],

where g is given by g(z, [zeta]) = 2[beta] - [zeta] + [2([zeta] - [beta])/z] ln(1 + z), z [member of] U, [zeta] [member of] [bar.U] The function q is convex and it is the best dominant.

Proof. Following the same steps as in the proof of Theorem 2.8 and considering p(z, [zeta]) = ([RI.sup.[alpha].sub.m,[lambda],l]f(z, [zeta])), the strong differential subordination (2.20) becomes

p(z, [zeta]) + [zp'.sub.z](z, [zeta]) [??][??] h(z, [zeta]) = [[zeta] + 2[beta] - [zeta])z/1 + z], z [member of] U, [zeta] [member of] [bar.U].

By using Lemma 1.1 for n = 1 and [gamma] = 1, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and this completes the proof.

REFERENCES

[1] Alina Alb Lupas: On a certain subclass of analytic functions defined by multiplier transformation and Ruscheweyh derivative, submitted.

[2] Alina Alb Lupas: A new comprehensive class of analytic functions defined by multiplier transformation, Math. Comput. Modelling, 54(2011), 2355-2362.

[3] Alina Alb Lupas: Certain special differential superordinations using multiplier transformation and Ruscheweyh derivative, J. Comput. Anal. Appl., 13(2011), No. 1, 108-115.

[4] A. Alb Lupas, G. I. Oros and Gh. Oros: On special strong differential subordinations using Salagean and Ruscheweyh operators, J. Comput. Anal. Appl., 14(2012), in printing.

[5] A. Alb Lupas and D. Breaz: A note on strong differential subordinations using Salagean operator and Ruscheweyh derivative, submitted.

[6] Alina Alb Lupas: On special strong differential subordinations using multiplier transformation, Appl. Math. Lett., 25(2012), 624-630.

[7] A. Alb Lupas, G. I. Oros and Gh. Oros: A note on special strong differential subordinations using multiplier transformation, J. Comput. Anal. Appl., 14(2012), in printing.

[8] Alina Alb Lupas: On special strong differential subordinations using a generalized Salagean operator and Ruscheweyh derivative, J. Concr. Appl. Math., 2012, in printing.

[9] A. Alb Lupas: A note on special strong differential subordinations using a generalized Salagean operator and Ruscheweyh derivative, submitted.

[10] J. A. Antonino and S. Romaguera: Strong differential subordination to Briot-Bouquet differential equations, J. Diff. Equ., 114(1994), 101-105.

[11] G. I. Oros: On a new strong differential subordination, (to appear).

[12] G. I. Oros and Gh. Oros: Strong differential subordination,Turk. J. Math., 33(2009), 249-257.

[13] St. Ruscheweyh: New criteria for univalent functions, Proc. Amer. Math. Soc., 49(1975), 109-115.

University of Oradea, Faculty of Sciences

Department of Mathematics and Computer Science