# 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  the notion of strong differential subordinations, which was developed by G. I. Oros and Gh. Oros in , .

Definition 1.1 (). 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 (). 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 (). 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 (). 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  and the multiplier transformation  to the new class of analytic functions [A.sup.*.sub.n[zeta]] introduced in .

Definition 1.2 (). 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 (). 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 (). 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 (). 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 ,  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 , .

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 ,  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 , .

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

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

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

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

 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.

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

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

 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.

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

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

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

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

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

 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

Author: Printer friendly Cite/link Email Feedback Lupas, Alina Alb Journal of Advanced Mathematical Studies Report 4EXRO Jan 1, 2012 4332 Harmonicity and submanifold maps. Suzuki contraction theorem on a 2-metric space. Derivatives (Mathematics) Differential equations Operator theory Transformations (Mathematics)