GHEORGHE OROS, ROXANA S¸ENDRUT¸ IU and GEORGIA IRINA OROS
Communicated by the former editorial board
In this paper we study the special case of first order strong differential superor- dinations. The notion of strong differential superordination is obtained as a dual concept of strong differential subordination, which is developed from the classic notion of differential subordination.
AMS 2010 Subject Classification: 30C80, 30C45, 30A20.
Key words: differential subordination, strong differential subordination, differen- tial superordination, strong differential superordination, univalent function, subordinant, best subordinant, dominant, best dominant.
1.INTRODUCTION AND PRELIMINARIES
Let U denote the unit disc of the complex plane, U tz PC: |z| 1u andU tzPC: |z| ¤1u.LetHpUUqdenote the class of analytic functions inU U. In [8] the author has defined the classes
Hζra, ns tf PHpUUq: fpz, ζq a anpζqzn an 1pζqzn 1 , zPU, ζ PUu
withakpζq holomorphic functions in U,k¥n,
HζupUq tf PHζra, ns: fp, ζqunivalent in U for all ζ PUu,
Aζn tf PHζra, ns: fpz, ζq z a2pζqz2 anpζqzn , zPU, ζ PUu withAζ1Aζ. Let
Sζ
"
f PAζ : Rezf1pz, ζq
fpz, ζq ¡0, zPU, for all ζ PU
*
denote the class of starlike functions in U U,
Kζ
"
f PAζ : Rezf2pz, ζq
f1pz, ζq 1¡0, z PU, for all ζ PU
*
denote the class of normalized convex functions in U U.
MATH. REPORTS15(65),2(2013), 115–124
Remark 1.1 The notion of differential superordination was introduced in [4] by S.S. Miller and P.T. Mocanu as a dual concept of differential subordina- tion [2] and developed in [3]. The concept of strong differential subordination was introduced in [1] by J.A. Antonino and S. Romaguera and was developed in [7, 8]. In [5] the author introduces the dual concept of strong differential superordination, and it was developed in [9].
In order to prove our main results we use the following definitions and lemmas:
Let Ω be any set in the complex plane C, let p be analytic in U and let ψpr, s, t;z, ζq : C3U U Ñ C. In a series of articles S.S. Miller and P.T.
Mocanu have determined properties of functions p that satisfy the differential subordinationtψpppzq, zp1pzq, z2p2pzq;zq: zPUu Ω.
In [4] S.S. Miller and P.T. Mocanu considered the dual problem of de- termining properties of functionspthat satisfy the differential superordination Ω tψpppzq, zp1pzq, z2p2pzq;zq: zPUu.
In [7] the authors have determined properties of functions ppz, ζq analytic in U U that satisfy the strong differential subordination tψpppz, ζq, zp1pz, ζq, z2p2pz, ζq;z, ζq: zPU, ζ PUu Ωζ.
In [5] the author has considered the dual problem of determining prop- erties of functions ppz, ζq that satisfy the strong differential superordination Ωζ tψpppz, ζq, zp1pz, ζq, z2p2pz, ζq;z, ζq: zPU, ζ PUu.
Let Ωζ and ∆ζ be any sets inC, let ppz, ζqbe analytic in U U and let ϕpr, s, t;z, ζq:C3UU ÑC. In that article they considered conditions on Ωζ, ∆ζ and ϕ, for which the following implication holds:
(1.1) Ωζ ψpppz, ζq, zp1pz, ζq, z2p2pz, ζq;z, ζq: zPU, ζ PU implies ∆ζ ppUUq.
If Ωζ and ∆ζ are simply connected domains with Ωζ C, ∆ζ Cthen there is a conformal mapping hpz, ζq of U U onto Ωζ such that hp0, ζq ϕppp0, ζq,0,0; 0, ζq and there is a conformal mappingqpz, ζq of UU onto ∆ζ such that qp0, ζq pp0, ζq, for allζ PU.
If ϕpppz, ζq, zp1pz, ζq, z2p2pz, ζq;z, ζq is univalent in U for all ζ PU, then 1.1 can be rewritten as
(1.2) hpz, ζq pppz, ζq, zp1pz, ζq, z2p2pz, ζq;z, ζq
implies qpz, ζq ppz, ζq, zPU,ζ PU. This implication also has meaning if hpz, ζq and qpz, ζq are analytic and not necessarily univalent.
Definition 1.1 ([7]). Let Ωζ be a set in C, qp, ζq P Ωζ and n be a posi- tive integer. The class of admissible functions ψnrΩζ, qp, ζqs consists of those
functions ψ:C3U U ÑCthat satisfy the admissibility condition:
(A) ψpr, s, t;z, ζq RΩζ,
whenever
r qpζ, ζq, smζq1pζ, ζq, Re t
s 1
¥mRe
ζq2pζ, ζq q1pζ, ζq 1
, zPU,ζ P BUzEpqq,ζ PU and m¥n.
We writeψ1rΩζ, qp, ζqsasψrΩζ, qp, ζqs.
Definition 1.2 ([9]). Letϕ:C3U U ÑC and let hpz, ζq be analytic in U U. If ppz, ζq and ϕpppz, ζq, zp1pz, ζq, z2p2pz, ζq;z, ζq are univalent in U for all ζ PU and satisfy the (second-order) strong differential superordination (1.3) hpz, ζq ϕpppz, ζq, zp1pz, ζq, z2p2pz, ζq;z, ζq
thenppz, ζq is called a solution of the strong differential superordination.
An analytic functionqpz, ζqis called a subordinant of the solutions of the strong differential superordination, or more simply a subordinant, ifqpz, ζq ppz, ζq for allppz, ζq satisfying 1.3.
An univalent subordinant rqpz, ζq that satisfies qpz, ζq rqpz, ζq for all subordinants qpz, ζq of 1.3 is said to be the best subordinant. Note that the best subordinant is unique up to a rotation of U.
Related to the strong superordination 1.2, three problems can be stated:
Problem 1. Given analytic functions hpz, ζq and qpz, ζq, find a class of admissible functions φrhpz, ζq, qpz, ζqssuch that 1.2 holds.
Problem 2. Given the strong differential superordination in 1.2, find a subordinantqpz, ζq. Moreover, find the best subordinant.
Problem 3. Given ϕpz, ζq and subordinant qpz, ζq, find the largest class of analytic functions hpz, ζq such that 1.2 holds.
Definition 1.3 ([8]).We denote by Qthe set of functions qpz, ζq that are analytic and injective, as function of z on U Epqq where Epqq tζ P BU :
zÑζlimqpz, ζq 8u.The subclass of Q for whichfp0q ais denoted byQpaq. Lemma A ([3, T. 2.6.h, p. 67]). If Lγ:AζÑAζ is the integral operator defined by Lγrfpzq, ζs Fpz, ζq γ 1
zγ
»z
0
fpt, ζqtγ1dt andRe γ ¥0, then (i) LγrSζs Sζ,
(ii) LγrKζs Kζ.
In paper [6], using the definitions given by Pommerenke [10] and Miller and Mocanu [3], the author introduced the notion of strong subordination (or Loewner) chain as follows:
Definition 1.4 ([6]). The function L : U U r0,8q Ñ C is a strong subordination (or a Loewner) chain if Lpz, ζ;tq is analytic and univalent inU forζ PU,t¥0,Lpz, ζ;tq is continuously differentiable function ofton r0,8q for all zPU,ζ PU, and Lpz, ζ;sq Lpz, ζ;tq where 0¤s¤t.
The following lemma provides a sufficient condition for Lpz, ζ;tq to be a strong subordination chain and it was obtained by adapting the results formed in [10, p. 159], [3, p. 4].
LemmaB ([6]). The function Lpz, ζ;tq a1pζ, tqz a2pζ, tqz2 . . . with a1pζ, tq 0 for ζ PU, t¥0 and lim
tÑ8|a1pζ, tq| 8, is a strong subordination chain if
Re zBLpz, ζ;tq{Bz
BLpz, ζ;tq{Bt ¡0, zPU, ζ PU , t¥0.
LemmaC ([9, Th. 2]). Let hp, ζqbe analytic inUU,qp, ζq PHζra, ns, ϕPC2U U ÑC and suppose that
(A) ϕpqpz, ζq, tzq1pz, ζq;ζ, ζq PhpUUq, for z P U, ζ P BU, ζ P U and 0 t ¤ 1
n ¤ 1. If pp, ζq P Qpaq and ϕpppz, ζq, zp1pz, ζq;z, ζq is univalent in U, for allζ PU then
hpz, ζq ϕpppz, ζq, zp1pz, ζq;z, ζq ùñ pz, ζq ppz, ζq.
Furthermore, if ϕpppz, ζq, zp1pz, ζq;z, ζq hpz, ζq, z P U, ζ P U has a univalent solution qp, ζq PQpaq, then qp, ζq is the best subordinant.
2. MAIN RESULTS
Theorem2.1. Leth1pz, ζqbe convex inU, for allζ PU withh1p0, ζq a, γ 0 withRe γ ¡0 andpPHζra,1s XQ. Ifppz, ζq zp1pz, ζq
γ is univalent in U, for all ζPU,
(2.1) h1pz, ζq ppz, ζq zp1pz, ζq
γ , and
(2.2) q1pz, ζq γ zγ
»z
0
h1pt, ζqtγ1dt, then q1pz, ζq ppz, ζq, zPU, ζ PU .
The function q1pz, ζq is convex and is the best subordinant.
Proof. Let ϕ:C2 ÑC, ϕpr, sq r s
γ, r PC, sPC,γ PC, Reγ ¥0.
For rppz, ζq,szp1pz, ζq,zPU, ζ PU we have
(2.3) ϕpppz, ζq, zp1pz, ζqq ppz, ζq zp1pz, ζq
γ .
Then, 2.1 becomes
(2.4) h1pz, ζq ϕpppz, ζq, zppz, ζqq, zPU, ζ PU .
Since h1pz, ζq is convex in U for all ζ PU and Re γ ¥0, using condition (ii) from LEMMA A, we deduce thatq1pz, ζq is convex and univalent inU for all ζ PU. Differentiating 2.2, we obtain
(2.5) h1pz, ζq q1pz, ζq zq11pz, ζq
γ ϕpq1pz, ζq, zq11pz, ζqq.
Since q1p, ζq is the univalent solution of the differential equation 2.5 as- sociated with strong differential superordination 2.1, we can prove that it is the best subordinant of 2.1 by applying Lemma C. If we let
(2.6) Jrz, ζ;ts ϕpq1pz, ζq, tzq11pz, ζqq q1pz, ζq tzq1pz, ζq γ , then Jrz, ζ;ts P h1pU Uq, for z P U, ζ P U and t P p0,1s, which is equivalent to ϕ P φrh1pz, ζq, q1pz, ζqs. From the well-known Hallenbeck and Ruschweyh Lemma [3, p. 71] we have
(j)ppz, ζq zp1pz, ζq
γ hpz, ζq,zPU,ζPU, wherehpz, ζqis convex inU for all ζ PU,hp0, ζq a,γ 0, Re γ ¥0. They showed that ifppz, ζq PHζra,1s satisfies (j), then
(jj) ppz, ζq qpz, ζq hpz, ζq, z P U, ζ P U, where qpz, ζq γ
zγ
»z
0
hpt, ζqtγ1dt, z P U, ζ PU . The function qpz, ζq is a convex function and is the best dominant of (j). From 2.5 and 2.6 we obtain
(2.7) Jrz, ζ;ts p1tqq1pz, ζq th1pz, ζq, zPU, ζPU , tP p0,1s. Since h1 is a convex function, from 2.5, we see that (j) is satisfied with ppz, ζq and hpz, ζq replaced by q1pz, ζq and respectively h1pz, ζq. Hence from (jj) we obtainq1pz, ζq h1pz, ζq,zPU,ζ PU, andq1is convex and univalent.
Since h1pUUq is a convex domain andtP p0,1swe conclude thatJrz, ζ;ts P h1pU Uq, which proves that q1pz, ζq is the best subordinant. l
Remark 1.2. Without loss of generality, we can assume thath1 andq1are analytic and univalent inU, and q11pζ, ζq 0 for|ζ| 1. If not, then we could replace h1pz, ζq by h1pρz, ζq and q1pz, ζq by q1pρz, ζq where 0 ρ 1. These
new functions would then have the desired properties and we would prove the theorem by using Lemma C and then letting ρÑ1.
We can combine this last theorem with the above mentioned Hallenbeck and Ruscheweyh result to obtain the following differential sandwich theorem.
Corollary 2.1. Let h1pz, ζq and h2pz, ζq be convex in U for all ζ PU, with h1p0, ζq h2p0, ζq a. Let γ 0, with Re γ ¥ 0, and let the functions qipz, ζq, zPU, ζ PU be defined by
(2.8) qipz, ζq γ zγ
»z
0
hipt, ζqtγ1dt, f or i1,2.
If ppz, ζq PHζra,1s XQ and ppz, ζq zp1pz, ζq
γ is univalent in U, for all ζ PU, then
(2.9) h1pz, ζq ppz, ζq zp1pz, ζq
γ h2pz, ζq, implies
q1pz, ζq ppz, ζq q2pz, ζq, zPU, ζ PU .
The functions q1pz, ζq and q2pz, ζq are convex and they are respectively the best subordinant and best dominant.
If we set fpz, ζq ppz, ζq zp1pz, ζq
γ then 2.9 can be expressed as the following sandwich theorem involving subordination preserving integral oper- ators.
Corollary 2.2. Leth1pz, ζqandh2pz, ζq, convex inU, for allζ PU and fpz, ζq be univalent in U, for all ζ PU with h1p0, ζq h2p0, ζq fp0, ζq. Let γ 0 with Re γ ¥0. If
h1pz, ζq fpz, ζq h2pz, ζq, zPU, ζ PU , then
γ zγ
»z
0
h1pt, ζqtγ1dt γ zγ
»z
0
fpt, ζqtγ1dt γ zγ
»z
0
h2pt, ζqtγ1dt, where the middle integral is univalent.
Theorem 2.2. Let qpz, ζq be convex in U, for allζ PU and lethpz, ζq be defined by
(2.10) qpz, ζq zq1pz, ζq
γ hpz, ζq, zPU, ζ PU , withRe γ ¡0.
If ppz, ζq PHζra,1sXQ,ppz, ζq zp1pz, ζq
γ is univalent inU, for allζ PU and satisfy
(2.11) hpz, ζq ppz, ζq zppz, ζq
γ , zPU, ζ PU , then qpz, ζq ppz, ζq,
where
qpz, ζq γ zγ
»z
0
hpt, ζqtγ1dt, zPU, ζ PU . The function q is the best subordinant.
Proof. Let ϕ:C2 ÑC, ϕpr, sq r s
γ. If r qpz, ζq, szq1pz, ζq, we let Lpz, ζ;tq ϕpqpz, ζq, tzq1pz, ζqq qpz, ζq tzq1pz, ζq
γ , z PU, ζ PU , t¥0.
A simple calculation leads to BLpz, ζ;tq
Bz q1pz, ζq tq1pz, ζq tzq2pz, ζq γ
and BLpz, ζ;tq
Bt zg1pz, ζq
γ , z P U, ζ P U . Then Re
zBLpz, ζ;tq{Bz BLpz, ζ;tq{Bt
Re
zqpz, ζqγ tq1pz, tq t γq1pz, ζq zq1pz, ζq
γ
Re
γ t
1 zq2pz, ζq q1pz, ζq
¡0, zPU. Since
Lpz, ζ;tq satisfies the conditions of Lemma B, we deduce that Lpz, ζ;tq is a subordination chain. For t 1, Lpz, ζ,1q ϕpqpz, ζq, zqpz, ζqq qpz, ζq
zqpz, ζq
γ hpz, ζq.SinceLpz, ζ;tqis a subordination chain we haveLpz, ζ;tq Lpz, ζ,1q,or equivalently,
(2.12) qpz, ζq tzq1pz, ζq
γ qpz, ζq zqpz, ζq
γ hpz, ζq. From 2.11 and 2.12 we have qpz, ζq tzq1pz, ζq
γ ppz, ζq zppz, ζq
γ .
Using Lemma C, we conclude that qpz, ζq ppz, ζq, z P U, ζ P U. Fur- thermore, sinceq is a univalent solution of 2.10, it is also the best subordinant of 2.11. l
Remark 1.3 This last theorem is an example of solution to Problem 3 referred to in the introduction.
The remaining two theorems are examples of solutions to Problem 2.
They involve differential superordinations for which the subordinate function h is a starlike function.
Theorem2.3. Lethpz, ζqbe starlike inU, for allζ PU, withhp0, ζq 0.
If ppz, ζq PHζr0,1s XQ and zp1pz, ζq is univalent in U, for allζ PU, then (2.13) hpz, ζq zp1pz, ζq
implies
qpz, ζq ppz, ζq, zPU, ζ PU , where
(2.14) qpz, ζq
»z
0
hpt, ζqtγ1dt.
The function q is convex and is the best subordinant.
Proof. If we letϕps, ζq s, then 2.13 becomes (2.15) hpz, ζq ϕpzp1pz, ζqq, zPU, ζPU .
Differentiating 2.14 we obtain
(2.16) zq1pz, ζq hpz, ζq, zPU, ζ PU .
Since the function qp, ζq is a solution of 2.16, and hp, ζq is starlike, we deduce that qp, ζq is convex and univalent. As in the previous theorems we can assume that hpz, ζq and qpz, ζqare analytic in UU, and univalent inU for allζ PU, andq1pζ, ζq 0 for|ζ| 1,ζ PU. The conclusion of this theorem follows from Lemma C if we show that admissibility condition (A) is satisfied.
This follows immediately since hpU Uq is a starlike domain, for all ζ P U and ϕptzq1pz, ζqq tzq1pzq thpz, ζq hpUϕq,zPU, ζ PU and 0 t¤1.
Applying Lemma B we conclude thatqp, ζq is the best subordinant. l Example 2.1.If we lethpz, ζq zζ,zPU,ζ PU, then
Re zh1pz, ζq
hpz, ζq Re zζ
zζ 1¡0,
hence, hpz, ζq is starlike in U, for all ζ P U. If ppz, ζq P Hζr0,1s X Q and zp1pz, ζq is univalent in U for all ζ PU thenzζ zp1pz, ζq implies
qpz, ζq zζ ppz, ζq, zPU, ζ PU , where
qpz, ζq
»z
0
ptζqt1dtζ
»z
0
dtzζ, zPU, ζ PU .
Remark 1.4. There is a corresponding result of Theorem 2.3 for strong differential subordinations of the form zp1pz, ζq hpz, ζq,z P U,ζ P U due to T.J. Suffridge [3, p. 76]. If we combine that result with Theorem 2.3, we obtain the following sandwich result.
Corollary 2.3. Let h1p, ζq and h2p, ζq be starlike in U for all ζ P U, with h1p0, ζq h2p0, ζq 0, and let the functions qipz, ζq be defined by qipz, ζq ³z
0hipt, ζqtγ1dt, for i 1,2. If p P Hζr0,1s XQ and zp1pz, ζq is univalent in U, for all ζ PU, then
h1pz, ζq zp1pz, ζq h2pz, ζq implies
q1pz, ζq ppz, ζq q2pz, ζq, zPU, ζ PU .
The functions q1p, ζqand q2p, ζqare convex and they are respectively the best subordinant and best dominant.
If we setfpz, ζq zppz, ζq, then this last Corollary 2.3 can be expressed as the following sandwich theorem involving a subordination preserving integral operator.
Corollary 2.4. Let h1p, ζq and h2p, ζq be starlike in U for all ζ P U with h1p0, ζq h2p0, ζq 0. If
h1pz, ζq fpz, ζq h2pz, ζq,
then »z
0
h1pt, ζqt1dt
»z
0
fpt, ζqt1dt
»z
0
h2pt, ζqt1dt, where the middle integral is univalent.
Example 2.2.If we leth2pz, ζq z2ζ zζ,zPU,ζ PU, then we have Re zh12pz, ζq
h2pz, ζq Re 2z 1
z 1 1 Re z
z 1 3
2 ¡0.
Using the result from Example 2.1 and Corollary 2.3 we have:
Ifppz, ζq PHζr0,1s XQand zp1pz, ζqis univalent inU, for allζ PU, then zζ zp1pz, ζq z2ζ zζ
implies »z
0
ptζqt1dt ppz, ζq
»z
0
pt2ζ tζqt1dt and zζ ppz, ζq z2
2ζ zζ, where q1pz, ζq zζ is a convex function and best subordinant and q2pz, ζq z2
2 ζ zζ is a convex function and best dominant.
REFERENCES
[1] J.A. Antonino and S. Romaguera, Strong differential subordination to Briot-Bouquet differential equations. Journal of Differential Equations114(1994), 101–105.
[2] S.S. Miller and P.T. Mocanu,Differential subordinations and univalent functions. Michig.
Math. J.28(1981), 157–171.
[3] S.S. Miller and P.T. Mocanu,Differential subordinations. Theory and applications. Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 2000.
[4] S.S. Miller and P.T. Mocanu, Subordinant of differential superordinations. Complex Variables48(2003),10, 815–826.
[5] Georgia Irina Oros, Strong differential superordination. Acta Universitatis Apulensis, 19(2009), 110–116.
[6] Georgia Irina Oros, An application of the subordination chains. Fractional Calculus Applied Analysis13(2010),5, 521–530
[7] G. I. Oros and Gh. Oros,Strong differential subordination. Turkish Journal of Mathe- matics,33(2009), 249–257.
[8] Georgia Irina Oros,On a new strong differential subordination(to appear).
[9] Gheorghe Oros, Briot-Bouquet strong differential superordination and sandwich theo- rems. Math. Reports12(62) (2010),3,277–283.
[10] Ch. Pommerenke,Univalent Functions. Vanderhoeck and Ruprecht, G¨ottingen, 1975.
Received 2 May 2011 University of Oradea,
Faculty of Sciences, Str. Universit˘at¸ii, No.1, 410087
Oradea, Romania georgia oros ro@yahoo.co.uk
University of Oradea, Faculty of Environmental Protection,
Str. Universit˘at¸ii, No.1, 410087 Oradea, Romania roxana.sendrutiu@gmail.com
