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Harmonic mappings related to the m-fold starlike functions
Applied Mathematics and Computation 267 (2015) 805–809
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Harmonic mappings related to the m-fold starlike functions Melike Aydog˘an a,⇑, Yasßar Polatog˘lu b, Yasemin Kahramaner c _ Turkey Department of Mathematics, Isßık University, Mesßrutiyet Koyu, Sßile, Istanbul, _ _ Ku˘ltu˘r Universitesi, Istanbul, Turkey Department of Mathematics and Computer Science, Istanbul c _ _ Ticaret University, Istanbul, Turkey Department of Mathematics, Istanbul a
b
a r t i c l e
i n f o
Keywords: m-fold starlike functions Distortion theorem Growth theorem
a b s t r a c t In the present paper we will give some properties of the subclass of harmonic mappings which is related to m-fold starlike functions in the open unit disc D ¼ fzjjzj < 1g. Throughout this paper we restrict ourselves to the study of sense-preserving harmonic mappings. We also note that an elegant and complete treatment theory of the harmonic mapping is given in Durens monograph (Duren, 1983). The main aim of us to investigate some properties of the new class of us which represented as in the following form,
S HðmÞ ¼ where hðzÞ ¼ z þ
g 0 ðzÞ b1 pðzÞ; hðzÞ 2 S ðmÞ; pðzÞ 2 PðmÞ ; f ¼ hðzÞ þ gðzÞjf 2 SHðmÞ; 0 h ðzÞ P1
P mnþ1 gðzÞ ¼ 1 ; jb1 j < 1. n¼0 bmnþ1 z Crown Copyright Ó 2014 Published by Elsevier Inc. All rights reserved. mnþ1 , n¼1 amnþ1 z
1. Introduction Let X be the family of functions /ðzÞ which are analytic in D and satisfying the conditions /ð0Þ ¼ 0; j/ðzÞj < 1 for every z 2 D. Let PðmÞ denote the set of functions of the form pðzÞ ¼ 1 þ pm zm þ p2m z2m þ . . . for which RepðzÞ > 0 in D. For brevity we say that pðzÞ has m-fold symmetry. A function is called m-fold symmetric is its power series has the form [1,4,5].
sðzÞ ¼
1 X dmnþ1 zmnþ1
ð1:1Þ
n¼0
2pi 2pi This condition on the power series is equivalent to the condition s e m z ¼ e m sðzÞ for z 2 D. The class of such functions is ðmÞ denoted by S . We denote by S ðmÞ the families m-fold symmetric starlike functions. That is
0 s ðzÞ >0 sðzÞ 2 S ðmÞ () Re z sðzÞ
ð1:2Þ
[1,4,5]. Let s1 ðzÞ ¼ z þ d2 z2 þ . . . and s2 ðzÞ ¼ z þ e2 z2 þ . . . be analytic functions in D. If there exists a function /ðzÞ 2 X such that s1 ðzÞ ¼ s2 ð/ðzÞÞ for every z 2 D. Then we say that s1 ðzÞ is subordinate to s2 ðzÞ and we write s1 ðzÞ s2 ðzÞ. Specially if s2 ðzÞ is univalent in D, then s1 ðzÞ s2 ðzÞ if and only if s1 ðDÞ s2 ðDÞ and s1 ð0Þ ¼ s2 ð0Þ implies s1 ðDr Þ s2 ðDr Þ where Dr ¼ fzjjzj < r; 0 < r < 1g [1,4]. A planar harmonic mapping in the open unit disc D is a complex valued harmonic function f, which maps D onto the some planar domain f ðDÞ. Since D is a simply connected domain the mapping f has a canonical decomposition f ¼ hðzÞ þ gðzÞ, where hðzÞ and gðzÞ are analytic in D and have the following power series expansions, ⇑ Corresponding author. E-mail addresses: [email protected] (M. Aydog˘an), [email protected] (Y. Polatog˘lu), [email protected] (Y. Kahramaner). http://dx.doi.org/10.1016/j.amc.2014.10.016 0096-3003/Crown Copyright Ó 2014 Published by Elsevier Inc. All rights reserved.
M. Aydog˘an et al. / Applied Mathematics and Computation 267 (2015) 805–809
806
hðzÞ ¼ z þ
1 X an zn ;
gðzÞ ¼
n¼2
1 X b n zn n¼1
as usual, we call hðzÞ the analytic part and gðzÞ is the co-analytic part of f, respectively and let the class of such harmonic mappings is denoted by SH, (see [2]) proved in 1936 that the harmonic mapping f is locally univalent in D if and only if 0 its Jacobian J f ¼ ðjh ðzÞj2 jg 0 ðzÞj2 Þ is strictly positive univalent harmonic mapping in 0 in D. In view of this result, 0 locally the unit disc are either sense-reversing if jg 0 ðzÞj > h ðzÞ or sense-preserving if h ðzÞ > jg 0 ðzÞj in D. Throughout this paper we restrict ourselves to the study of sense-preserving harmonic mappings. We also note that an elegant and complete treatment theory of the harmonic mapping is given Duren’s monograph [2]. The main aim of this paper is to investigate the some properties of the following class.
S HðmÞ ¼
g 0 ðzÞ f ¼ hðzÞ þ gðzÞjf 2 SHðmÞ; 0 b1 pðzÞ; hðzÞ 2 S ðmÞ; pðzÞ 2 P ðmÞ ; h ðzÞ
where P P mnþ1 mnþ1 hðzÞ ¼ z þ 1 , gðzÞ ¼ 1 ; jb1 j < 1. and for this aim we will need the following lemmas. n¼1 amnþ1 z n¼0 bmnþ1 z Lemma 1.1 [4]. Let m P 1 be a fixed integer and assume that
/ðzÞ ¼ B2m z2m þ B2mþ1 z2mþ1 þ . . . ¼
1 X
Bn zn
n¼2m
satisfies the conditions of Schwarz Lemma. If 0 < r < 1, then
j/ðzÞj 6 r 2m
ð1:3Þ ih
ia 2m
If equality occurs in (1.3) for one point z0 ¼ re with 0 < r < 1, then /ðzÞ ¼ e z
and the equal sign holds in (1.3) for all z 2 D.
Lemma 1.2 [3]. Let /ðzÞ ¼ an zn þ anþ1 znþ1 þ . . .ðan – 0; n P 1Þ be analytic in D. If the maximum value of j/ðzÞj on the circle jzj ¼ r < 1 is attained at z ¼ z0 , then we have z0 /0 ðz0 Þ ¼ m/ðz0 Þ; m P n and every z 2 D. k
Lemma 1.3 [4]. Let pðzÞ be an element of pðmÞ , then PðDr Þ is the open disc with diameter end points w1 ¼ 1r and w2 ¼ w11 . 1þr k Lemma 1.4 [5]. Let hðzÞ ¼ z þ amþ1 zmþ1 þ a2mþ1 z2mþ1 þ . . . be a starlike function on D. Then
r 2
ð1 þ rm Þm
6 jhðzÞj 6
r 2
ð1 r m Þm
1
1
0 ð1 þ r m Þm 6 h ðzÞ 6 2 2 ð1 þ rm Þm ð1 r m Þm ð1 rm Þm
2. Main results
Theorem 2.1. Let f ¼ hðzÞ þ gðzÞ be an element of S HðmÞ, then
2 b1 ð1 r 2m Þ ð1 jb1 jÞ r m ; 6 wðzÞ 2 2 2m 1 jb1 j r2m 1 jb 1 j r
ð2:1Þ
where wðzÞ is the second dilatation of f. Proof. Since f ¼ hðzÞ þ gðzÞ 2 S HðmÞ then we have,
wðzÞ ¼
g 0 ðzÞ b1 þ ðm þ 1Þbmþ1 zm þ ð2m þ 1Þb2mþ1 z2m þ ð3m þ 1Þb3mþ1 z3m þ . . . ¼ 0 1 þ ðm þ 1Þamþ1 zm þ ð2m þ 1Þa2mþ1 z2m þ ð3m þ 1Þa3mþ1 z3m þ . . . h ðzÞ
thus,
/ðzÞ ¼
wðzÞ wð0Þ 1 wð0ÞwðzÞ
¼ ½ðm þ 1Þbmþ1 b1 ðm þ 1Þamþ1 zm þ . . . ¼
wðzÞ b1 1 b1 wðzÞ
M. Aydog˘an et al. / Applied Mathematics and Computation 267 (2015) 805–809
807
Therefore /ðzÞ satisfies the conditions of Schwarz Lemma. Using Lemma 1.1, we can write
2 wðzÞ b a1 ð1 r2m Þ a2 ð1 r2m Þ jb1 j r2m 1 m m u2 v þ ¼0 6 jzj () jwðzÞ b1 j 6 jzj 1 b1 wðzÞ ) u2 þ v 2 2 2 2 2 1 b1 wðzÞ 1 jb1 j r 2m 1 jb1 j r 2m 1 jb1 j r 2m Therefore we have,
a1 ð1 r2m Þ a2 ð1 r2m Þ
CðrÞ ¼
RðrÞ ¼
2
;
!
2
1 jb1 j r 2m 1 jb1 j r2m
;
2 1 jb 1 j r m 2
1 jb1 j r2m
:
which shows that this theorem is true.
h
Corollary 2.2. Let f ¼ hðzÞ þ gðzÞ be an element of S HðmÞ, then
2 1 jb1 j ð1 r 2m Þ ð1 þ jb1 jr m Þ
2
2 1 jb1 j B ð1 r 2m Þ 6 1 jwðzÞj2 6 2 ð1 jb1 jr m Þ
ð1 þ jb1 jÞð1 r m Þ ð1 þ jb1 jÞð1 þ r m Þ 6 ð1 þ jwðzÞjÞ 6 m 1 jb1 jr 1 þ jb1 jr m ð1 jb1 jÞð1 r m Þ ð1 jb1 jÞð1 þ r m Þ 6 ð1 jwðzÞjÞ 6 m 1 þ jb1 jr 1 jb1 jr m Proof. This Corollary is a simple consequence of Theorem 2.1.
h
Theorem 2.3. Let f ¼ hðzÞ þ gðzÞ be an element of S HðmÞ, then
gðzÞ 1 þ zm b1 pðzÞ ¼ b1 hðzÞ 1 zm
ð2:2Þ
where pðzÞ 2 Pm . Proof. Since f ¼ hðzÞ þ gðzÞ be an element of S HðmÞ, then using Theorem 2.1 we can write
wðDr Þ ¼
g 0 ðzÞ g 0 ðzÞ b1 ð1 þ r 2m Þ 2jb1 jrm : 6 j 0 0 1 r 2m 1 r 2m h ðzÞ h ðzÞ
On the other hand, since hðzÞ is an element of S ðmÞ, the value of
ð2:3Þ
hðzÞ zh0 ðzÞ
at a point z0 on the circle jzj ¼ r is,
2m
hðz0 Þ 1r : ¼ 0 z0 h ðz0 Þ ð1 þ r 2m Þ þ 2eih r m
ð2:4Þ
Now we define the function
gðzÞ 1 þ ð/ðzÞÞm ¼ b1 : hðzÞ 1 ð/ðzÞÞm
ð2:5Þ
Then
gðzÞ b1 z þ bmþ1 zmþ1 þ b2mþ1 z2mþ1 þ . . . b1 þ bmþ1 zm þ b2mþ1 z2m þ . . . gðzÞ 1 þ ð/ðzÞÞm ¼ ) ¼ b1 ¼ b1 ¼ mþ1 2mþ1 m 2m hðzÞ z þ amþ1 z 1 þ amþ1 z þ a2mþ1 z þ . . . hðzÞ z¼0 þ a2mþ1 z þ ... 1 ð/ðzÞÞm ) 1 ð/ð0ÞÞm ¼ 1 þ ð/ð0ÞÞm ) /ð0Þ ¼ 0;
/ðzÞ is analytic. We need to show that j/ðzÞj < 1 for all z 2 D. Assume to the contrary that there exists a z0 2 D such that j/ðz0 Þj ¼ 1. If we take derivative of (2.5) and after simple calculations we get
g 0 ðzÞ 1 þ ð/ðzÞÞm 2/0 ðzÞð/ðzÞÞm1 hðzÞ ¼ b1 m 2b1 m 0 0 2 1 ð/ðzÞÞ h ðzÞ ð1 ð/ðzÞÞm Þ zh ðzÞ
ð2:6Þ
M. Aydog˘an et al. / Applied Mathematics and Computation 267 (2015) 805–809
808
Considering (2.3), (2.4) and (2.6) and Lemma 1.2 together, then we obtain
g 0 ðz0 Þ 1 þ ð/ðz0 ÞÞm 2mkð/ðz0 ÞÞm 1 r2m wðz0 Þ ¼ 0 ¼ b1 m m 2 ð1 þ r 2m Þ þ 2eih r m 1 ð/ðz0 ÞÞ h ðz0 Þ ½1 ð/ðz0 ÞÞ
! R wðDr Þ
But this is a contradiction, therefore j/ðzÞj < 1 for all z 2 D. Thus, for a function f ¼ hðzÞ þ gðzÞ in S HðmÞ we have
gðzÞ 1 þ zm b1 pðzÞ ¼ b1 : hðzÞ 1 zm
Corollary 2.4. If f ¼ hðzÞ þ gðzÞ be an element of S HðmÞ, then 1
1
ðjb1 j rm Þð1 r m Þm ð1 jb1 jr m Þð1 þ r m Þ
3 m
r ðjb1 j r m Þ ð1 jb1
jr m Þð1
þ
2
r m Þm
6 jg 0 ðzÞj 6
6 jgðzÞj 6
ðjb1 j þ r m Þð1 þ rm Þm
ð2:7Þ
3
ð1 þ jb1 jr m Þð1 rm Þm r ðjb1 j þ rm Þ
ð2:8Þ
2
ð1 þ jb1 jr m Þð1 r m Þm
Proof. Using Theorems 2.1 and 2.3, then we can write
0 jb 1 j r m jb1 j þ rm 0 h ðzÞ 6 jg 0 ðzÞj 6 h ðzÞ m 1 jb1 jr 1 þ jb1 jrm jhðzÞj
ð2:9Þ
jb 1 j r m jb1 j þ r m 6 jgðzÞj 6 jhðzÞj m 1 jb1 jr 1 þ jb1 jr m
ð2:10Þ
If we use Theorem 1.4 in (2.9) and (2.10), then we obtain (2.7) and (2.8). h Corollary 2.5. If f ¼ hðzÞ þ gðzÞ be an element of S HðmÞ, then
1 jb1 j
2
2þm m
ð1 rm Þ
2
ð1 þ jb1 jr m Þ ð1 þ rm Þ 6m m
6 Jf 6
1 jb 1 j
2
2þm m
ð1 þ rm Þ
ð2:11Þ
2
ð1 jb1 jr m Þ ð1 r m Þ 6m m
Proof. Since
0 2 0 2 0 2 2 2 J f ¼ h ðzÞ jg 0 ðzÞj ¼ h ðzÞ jhðzÞwðzÞj ¼ h ðzÞ ð1 jwðzÞj2 Þ
ð2:12Þ
Using Theorem 1.4 and Corollary 2.2 in (2.12) we get (2.11). h Corollary 2.6. If f ¼ hðzÞ þ gðzÞ be an element of S HðmÞ, then
ð1 jb1 jÞ
Z
1
ð1 r m Þ
mþ1 m
dr
3
0
ð1 þ rm Þm ð1 þ jb1 jrm Þ
6 jf j 6 ð1 þ jb1 jÞ
Z
1
ð1 þ r m Þ
mþ1 m
dr
3
0
ð1 r m Þm ð1 þ jb1 jr m Þ
ð2:13Þ
Proof. Since
0 0 0 0 ðh ðzÞ jg 0 ðzÞjÞjdzj 6 jdf j 6 ðh ðzÞ þ jg 0 ðzÞjÞjdzj ) h ðzÞð1 jwðzÞjÞjdzj 6 df 6 h ðzÞð1 þ jwðzÞjÞjdzj using Theorem 1.4 and Corollary 2.2 in (2.14) we obtain (2.13).
ð2:14Þ
h
Corollary 2.7. If f ¼ hðzÞ þ gðzÞ be an element of S HðmÞ, then
ð1 jb1 jÞ
sin mp cos mp
Z 0
1
mþ1 p Z 1 sin m ð1 þ rm Þ m dr 6 hðzÞ gðzÞ 6 ð1 þ jb1 jÞ 3 p cos m 0 ð1 r m Þm ð1 jb1 jr m Þ ð1 þ r m Þ ð1 þ jb1 jrm Þ
ð1 r m Þ
mþ1 m
dr
3 m
Proof. Since f ¼ hðzÞ þ gðzÞ be an element of S HðmÞ, then we have
ð2:15Þ
M. Aydog˘an et al. / Applied Mathematics and Computation 267 (2015) 805–809 2p i
2pi
2pi
hðe m zÞ ¼ e m hðzÞ;
2pi
2pi
2pi
2pi
2pi
2p i
809 2pi m
gðe m zÞ ¼ e m gðzÞ ) f ðzÞ ¼ hðe m zÞ þ gðe m zÞ ¼ e m hðzÞ þ e m gðzÞ ¼ e m hðzÞ þ e
gðzÞ )
1 sin mp f ðzÞ i cos mp
¼ ðhðzÞ gðzÞÞ In this step if we use Corollary 2.6, we obtain the desired result. h Corollary 2.8. If f ¼ hðzÞ þ gðzÞ be an element of S HðmÞ, then
u1 ¼ cos
2p 2p v2 u2 þ sin m m
2p 2p sin v2 ; u2 þ cos m m
v1 ¼
where hðzÞ ¼ u1 þ iv 1 ; gðzÞ ¼ u2 þ iv 2 Proof. Using the definition of S HðmÞ and Corollary 2.7 then we can write,
f ðzÞ ¼ hðzÞ þ gðzÞ i
sin mp f ðzÞ ¼ hðzÞ gðzÞ: cos mp
Therefore we have
hðzÞ ¼
1 p eim f ðzÞ 2 cos mp
gðzÞ ¼
1 p im p e f ðzÞ 2 cos m
These equalities shows that this corollary is true. Thus we can say that if f ðzÞ 2 S HðmÞ, then hðzÞ and gðzÞ are linear dependent. We consider, this property is invariant for symmetric harmonic mappings. It has not been proved, yet. h Corollary 2.9. If f ¼ hðzÞ þ gðzÞ be an element of S HðmÞ, since for m ¼ 2, the functions hðzÞ and gðzÞ becomes odd then we have, (i)
ð1 jb1 jÞ
Z
1
3
ð1 r 2 Þ3 dr 3
0
ð1 þ r 2 Þ2 ð1 þ jb1 jr 2 Þ
6 jf j 6 ð1 þ jb1 jÞ
Z
1
3
ð1 þ r 2 Þ2 dr 3
0
ð1 r 2 Þ2 ð1 þ jb1 jr 2 Þ
(ii)
1
jb1 j r2 ð1 r 2 Þ2 3
ð1 jb1 jr2 Þð1 þ r2 Þ2
0
6 jg ðzÞj 6
1
jb1 j þ r2 ð1 þ r 2 Þ2 3
ð1 þ jb1 jr2 Þð1 r 2 Þ2
(iii)
r jb 1 j r 2 r jb1 j þ r 2 6 j gðzÞ j 6 ð1 jb1 jr2 Þð1 þ r2 Þ ð1 þ jb1 jr 2 Þð1 r 2 Þ
Proof. This corollary is a simple consequence of Corollaries 2.4 and 2.6. h References [1] [2] [3] [4] [5]
P. Duren, Univalent Functions, Grundiehren der Mathematischen Wissenchafter, vol. 259, Springer-Verlag, New York, 1983. P. Duren, Harmonic Mappings in the Plane, Cambridge Tracts in Mathematics, vol. 156, Cambridge University Press, Cambridge UK, 2004. S. Fukui, K. Sakaguchi, An extension of a theorem of S. Ruscheweyh, Bull. Fac. Ed. Wakayama Univ. Nat. Sci. 29 (1980) 1–3. A.W. Goodman, Univalent Functions, vols. I and II, Mariner publishing Co., Tampa Fl, 1983. I. Graham, Gabriela Kohr, Geometric function theory in one and higher dimensions, Pure and Applied Mathematics, A Dekker series Monographs and Textbooks, (2003).
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Harmonic mappings related to the m-fold starlike functions Melike Aydog˘an a,⇑, Yasßar Polatog˘lu b, Yasemin Kahramaner c _ Turkey Department of Mathematics, Isßık University, Mesßrutiyet Koyu, Sßile, Istanbul, _ _ Ku˘ltu˘r Universitesi, Istanbul, Turkey Department of Mathematics and Computer Science, Istanbul c _ _ Ticaret University, Istanbul, Turkey Department of Mathematics, Istanbul a
b
a r t i c l e
i n f o
Keywords: m-fold starlike functions Distortion theorem Growth theorem
a b s t r a c t In the present paper we will give some properties of the subclass of harmonic mappings which is related to m-fold starlike functions in the open unit disc D ¼ fzjjzj < 1g. Throughout this paper we restrict ourselves to the study of sense-preserving harmonic mappings. We also note that an elegant and complete treatment theory of the harmonic mapping is given in Durens monograph (Duren, 1983). The main aim of us to investigate some properties of the new class of us which represented as in the following form,
S HðmÞ ¼ where hðzÞ ¼ z þ
g 0 ðzÞ b1 pðzÞ; hðzÞ 2 S ðmÞ; pðzÞ 2 PðmÞ ; f ¼ hðzÞ þ gðzÞjf 2 SHðmÞ; 0 h ðzÞ P1
P mnþ1 gðzÞ ¼ 1 ; jb1 j < 1. n¼0 bmnþ1 z Crown Copyright Ó 2014 Published by Elsevier Inc. All rights reserved. mnþ1 , n¼1 amnþ1 z
1. Introduction Let X be the family of functions /ðzÞ which are analytic in D and satisfying the conditions /ð0Þ ¼ 0; j/ðzÞj < 1 for every z 2 D. Let PðmÞ denote the set of functions of the form pðzÞ ¼ 1 þ pm zm þ p2m z2m þ . . . for which RepðzÞ > 0 in D. For brevity we say that pðzÞ has m-fold symmetry. A function is called m-fold symmetric is its power series has the form [1,4,5].
sðzÞ ¼
1 X dmnþ1 zmnþ1
ð1:1Þ
n¼0
2pi 2pi This condition on the power series is equivalent to the condition s e m z ¼ e m sðzÞ for z 2 D. The class of such functions is ðmÞ denoted by S . We denote by S ðmÞ the families m-fold symmetric starlike functions. That is
0 s ðzÞ >0 sðzÞ 2 S ðmÞ () Re z sðzÞ
ð1:2Þ
[1,4,5]. Let s1 ðzÞ ¼ z þ d2 z2 þ . . . and s2 ðzÞ ¼ z þ e2 z2 þ . . . be analytic functions in D. If there exists a function /ðzÞ 2 X such that s1 ðzÞ ¼ s2 ð/ðzÞÞ for every z 2 D. Then we say that s1 ðzÞ is subordinate to s2 ðzÞ and we write s1 ðzÞ s2 ðzÞ. Specially if s2 ðzÞ is univalent in D, then s1 ðzÞ s2 ðzÞ if and only if s1 ðDÞ s2 ðDÞ and s1 ð0Þ ¼ s2 ð0Þ implies s1 ðDr Þ s2 ðDr Þ where Dr ¼ fzjjzj < r; 0 < r < 1g [1,4]. A planar harmonic mapping in the open unit disc D is a complex valued harmonic function f, which maps D onto the some planar domain f ðDÞ. Since D is a simply connected domain the mapping f has a canonical decomposition f ¼ hðzÞ þ gðzÞ, where hðzÞ and gðzÞ are analytic in D and have the following power series expansions, ⇑ Corresponding author. E-mail addresses: [email protected] (M. Aydog˘an), [email protected] (Y. Polatog˘lu), [email protected] (Y. Kahramaner). http://dx.doi.org/10.1016/j.amc.2014.10.016 0096-3003/Crown Copyright Ó 2014 Published by Elsevier Inc. All rights reserved.
M. Aydog˘an et al. / Applied Mathematics and Computation 267 (2015) 805–809
806
hðzÞ ¼ z þ
1 X an zn ;
gðzÞ ¼
n¼2
1 X b n zn n¼1
as usual, we call hðzÞ the analytic part and gðzÞ is the co-analytic part of f, respectively and let the class of such harmonic mappings is denoted by SH, (see [2]) proved in 1936 that the harmonic mapping f is locally univalent in D if and only if 0 its Jacobian J f ¼ ðjh ðzÞj2 jg 0 ðzÞj2 Þ is strictly positive univalent harmonic mapping in 0 in D. In view of this result, 0 locally the unit disc are either sense-reversing if jg 0 ðzÞj > h ðzÞ or sense-preserving if h ðzÞ > jg 0 ðzÞj in D. Throughout this paper we restrict ourselves to the study of sense-preserving harmonic mappings. We also note that an elegant and complete treatment theory of the harmonic mapping is given Duren’s monograph [2]. The main aim of this paper is to investigate the some properties of the following class.
S HðmÞ ¼
g 0 ðzÞ f ¼ hðzÞ þ gðzÞjf 2 SHðmÞ; 0 b1 pðzÞ; hðzÞ 2 S ðmÞ; pðzÞ 2 P ðmÞ ; h ðzÞ
where P P mnþ1 mnþ1 hðzÞ ¼ z þ 1 , gðzÞ ¼ 1 ; jb1 j < 1. and for this aim we will need the following lemmas. n¼1 amnþ1 z n¼0 bmnþ1 z Lemma 1.1 [4]. Let m P 1 be a fixed integer and assume that
/ðzÞ ¼ B2m z2m þ B2mþ1 z2mþ1 þ . . . ¼
1 X
Bn zn
n¼2m
satisfies the conditions of Schwarz Lemma. If 0 < r < 1, then
j/ðzÞj 6 r 2m
ð1:3Þ ih
ia 2m
If equality occurs in (1.3) for one point z0 ¼ re with 0 < r < 1, then /ðzÞ ¼ e z
and the equal sign holds in (1.3) for all z 2 D.
Lemma 1.2 [3]. Let /ðzÞ ¼ an zn þ anþ1 znþ1 þ . . .ðan – 0; n P 1Þ be analytic in D. If the maximum value of j/ðzÞj on the circle jzj ¼ r < 1 is attained at z ¼ z0 , then we have z0 /0 ðz0 Þ ¼ m/ðz0 Þ; m P n and every z 2 D. k
Lemma 1.3 [4]. Let pðzÞ be an element of pðmÞ , then PðDr Þ is the open disc with diameter end points w1 ¼ 1r and w2 ¼ w11 . 1þr k Lemma 1.4 [5]. Let hðzÞ ¼ z þ amþ1 zmþ1 þ a2mþ1 z2mþ1 þ . . . be a starlike function on D. Then
r 2
ð1 þ rm Þm
6 jhðzÞj 6
r 2
ð1 r m Þm
1
1
0 ð1 þ r m Þm 6 h ðzÞ 6 2 2 ð1 þ rm Þm ð1 r m Þm ð1 rm Þm
2. Main results
Theorem 2.1. Let f ¼ hðzÞ þ gðzÞ be an element of S HðmÞ, then
2 b1 ð1 r 2m Þ ð1 jb1 jÞ r m ; 6 wðzÞ 2 2 2m 1 jb1 j r2m 1 jb 1 j r
ð2:1Þ
where wðzÞ is the second dilatation of f. Proof. Since f ¼ hðzÞ þ gðzÞ 2 S HðmÞ then we have,
wðzÞ ¼
g 0 ðzÞ b1 þ ðm þ 1Þbmþ1 zm þ ð2m þ 1Þb2mþ1 z2m þ ð3m þ 1Þb3mþ1 z3m þ . . . ¼ 0 1 þ ðm þ 1Þamþ1 zm þ ð2m þ 1Þa2mþ1 z2m þ ð3m þ 1Þa3mþ1 z3m þ . . . h ðzÞ
thus,
/ðzÞ ¼
wðzÞ wð0Þ 1 wð0ÞwðzÞ
¼ ½ðm þ 1Þbmþ1 b1 ðm þ 1Þamþ1 zm þ . . . ¼
wðzÞ b1 1 b1 wðzÞ
M. Aydog˘an et al. / Applied Mathematics and Computation 267 (2015) 805–809
807
Therefore /ðzÞ satisfies the conditions of Schwarz Lemma. Using Lemma 1.1, we can write
2 wðzÞ b a1 ð1 r2m Þ a2 ð1 r2m Þ jb1 j r2m 1 m m u2 v þ ¼0 6 jzj () jwðzÞ b1 j 6 jzj 1 b1 wðzÞ ) u2 þ v 2 2 2 2 2 1 b1 wðzÞ 1 jb1 j r 2m 1 jb1 j r 2m 1 jb1 j r 2m Therefore we have,
a1 ð1 r2m Þ a2 ð1 r2m Þ
CðrÞ ¼
RðrÞ ¼
2
;
!
2
1 jb1 j r 2m 1 jb1 j r2m
;
2 1 jb 1 j r m 2
1 jb1 j r2m
:
which shows that this theorem is true.
h
Corollary 2.2. Let f ¼ hðzÞ þ gðzÞ be an element of S HðmÞ, then
2 1 jb1 j ð1 r 2m Þ ð1 þ jb1 jr m Þ
2
2 1 jb1 j B ð1 r 2m Þ 6 1 jwðzÞj2 6 2 ð1 jb1 jr m Þ
ð1 þ jb1 jÞð1 r m Þ ð1 þ jb1 jÞð1 þ r m Þ 6 ð1 þ jwðzÞjÞ 6 m 1 jb1 jr 1 þ jb1 jr m ð1 jb1 jÞð1 r m Þ ð1 jb1 jÞð1 þ r m Þ 6 ð1 jwðzÞjÞ 6 m 1 þ jb1 jr 1 jb1 jr m Proof. This Corollary is a simple consequence of Theorem 2.1.
h
Theorem 2.3. Let f ¼ hðzÞ þ gðzÞ be an element of S HðmÞ, then
gðzÞ 1 þ zm b1 pðzÞ ¼ b1 hðzÞ 1 zm
ð2:2Þ
where pðzÞ 2 Pm . Proof. Since f ¼ hðzÞ þ gðzÞ be an element of S HðmÞ, then using Theorem 2.1 we can write
wðDr Þ ¼
g 0 ðzÞ g 0 ðzÞ b1 ð1 þ r 2m Þ 2jb1 jrm : 6 j 0 0 1 r 2m 1 r 2m h ðzÞ h ðzÞ
On the other hand, since hðzÞ is an element of S ðmÞ, the value of
ð2:3Þ
hðzÞ zh0 ðzÞ
at a point z0 on the circle jzj ¼ r is,
2m
hðz0 Þ 1r : ¼ 0 z0 h ðz0 Þ ð1 þ r 2m Þ þ 2eih r m
ð2:4Þ
Now we define the function
gðzÞ 1 þ ð/ðzÞÞm ¼ b1 : hðzÞ 1 ð/ðzÞÞm
ð2:5Þ
Then
gðzÞ b1 z þ bmþ1 zmþ1 þ b2mþ1 z2mþ1 þ . . . b1 þ bmþ1 zm þ b2mþ1 z2m þ . . . gðzÞ 1 þ ð/ðzÞÞm ¼ ) ¼ b1 ¼ b1 ¼ mþ1 2mþ1 m 2m hðzÞ z þ amþ1 z 1 þ amþ1 z þ a2mþ1 z þ . . . hðzÞ z¼0 þ a2mþ1 z þ ... 1 ð/ðzÞÞm ) 1 ð/ð0ÞÞm ¼ 1 þ ð/ð0ÞÞm ) /ð0Þ ¼ 0;
/ðzÞ is analytic. We need to show that j/ðzÞj < 1 for all z 2 D. Assume to the contrary that there exists a z0 2 D such that j/ðz0 Þj ¼ 1. If we take derivative of (2.5) and after simple calculations we get
g 0 ðzÞ 1 þ ð/ðzÞÞm 2/0 ðzÞð/ðzÞÞm1 hðzÞ ¼ b1 m 2b1 m 0 0 2 1 ð/ðzÞÞ h ðzÞ ð1 ð/ðzÞÞm Þ zh ðzÞ
ð2:6Þ
M. Aydog˘an et al. / Applied Mathematics and Computation 267 (2015) 805–809
808
Considering (2.3), (2.4) and (2.6) and Lemma 1.2 together, then we obtain
g 0 ðz0 Þ 1 þ ð/ðz0 ÞÞm 2mkð/ðz0 ÞÞm 1 r2m wðz0 Þ ¼ 0 ¼ b1 m m 2 ð1 þ r 2m Þ þ 2eih r m 1 ð/ðz0 ÞÞ h ðz0 Þ ½1 ð/ðz0 ÞÞ
! R wðDr Þ
But this is a contradiction, therefore j/ðzÞj < 1 for all z 2 D. Thus, for a function f ¼ hðzÞ þ gðzÞ in S HðmÞ we have
gðzÞ 1 þ zm b1 pðzÞ ¼ b1 : hðzÞ 1 zm
Corollary 2.4. If f ¼ hðzÞ þ gðzÞ be an element of S HðmÞ, then 1
1
ðjb1 j rm Þð1 r m Þm ð1 jb1 jr m Þð1 þ r m Þ
3 m
r ðjb1 j r m Þ ð1 jb1
jr m Þð1
þ
2
r m Þm
6 jg 0 ðzÞj 6
6 jgðzÞj 6
ðjb1 j þ r m Þð1 þ rm Þm
ð2:7Þ
3
ð1 þ jb1 jr m Þð1 rm Þm r ðjb1 j þ rm Þ
ð2:8Þ
2
ð1 þ jb1 jr m Þð1 r m Þm
Proof. Using Theorems 2.1 and 2.3, then we can write
0 jb 1 j r m jb1 j þ rm 0 h ðzÞ 6 jg 0 ðzÞj 6 h ðzÞ m 1 jb1 jr 1 þ jb1 jrm jhðzÞj
ð2:9Þ
jb 1 j r m jb1 j þ r m 6 jgðzÞj 6 jhðzÞj m 1 jb1 jr 1 þ jb1 jr m
ð2:10Þ
If we use Theorem 1.4 in (2.9) and (2.10), then we obtain (2.7) and (2.8). h Corollary 2.5. If f ¼ hðzÞ þ gðzÞ be an element of S HðmÞ, then
1 jb1 j
2
2þm m
ð1 rm Þ
2
ð1 þ jb1 jr m Þ ð1 þ rm Þ 6m m
6 Jf 6
1 jb 1 j
2
2þm m
ð1 þ rm Þ
ð2:11Þ
2
ð1 jb1 jr m Þ ð1 r m Þ 6m m
Proof. Since
0 2 0 2 0 2 2 2 J f ¼ h ðzÞ jg 0 ðzÞj ¼ h ðzÞ jhðzÞwðzÞj ¼ h ðzÞ ð1 jwðzÞj2 Þ
ð2:12Þ
Using Theorem 1.4 and Corollary 2.2 in (2.12) we get (2.11). h Corollary 2.6. If f ¼ hðzÞ þ gðzÞ be an element of S HðmÞ, then
ð1 jb1 jÞ
Z
1
ð1 r m Þ
mþ1 m
dr
3
0
ð1 þ rm Þm ð1 þ jb1 jrm Þ
6 jf j 6 ð1 þ jb1 jÞ
Z
1
ð1 þ r m Þ
mþ1 m
dr
3
0
ð1 r m Þm ð1 þ jb1 jr m Þ
ð2:13Þ
Proof. Since
0 0 0 0 ðh ðzÞ jg 0 ðzÞjÞjdzj 6 jdf j 6 ðh ðzÞ þ jg 0 ðzÞjÞjdzj ) h ðzÞð1 jwðzÞjÞjdzj 6 df 6 h ðzÞð1 þ jwðzÞjÞjdzj using Theorem 1.4 and Corollary 2.2 in (2.14) we obtain (2.13).
ð2:14Þ
h
Corollary 2.7. If f ¼ hðzÞ þ gðzÞ be an element of S HðmÞ, then
ð1 jb1 jÞ
sin mp cos mp
Z 0
1
mþ1 p Z 1 sin m ð1 þ rm Þ m dr 6 hðzÞ gðzÞ 6 ð1 þ jb1 jÞ 3 p cos m 0 ð1 r m Þm ð1 jb1 jr m Þ ð1 þ r m Þ ð1 þ jb1 jrm Þ
ð1 r m Þ
mþ1 m
dr
3 m
Proof. Since f ¼ hðzÞ þ gðzÞ be an element of S HðmÞ, then we have
ð2:15Þ
M. Aydog˘an et al. / Applied Mathematics and Computation 267 (2015) 805–809 2p i
2pi
2pi
hðe m zÞ ¼ e m hðzÞ;
2pi
2pi
2pi
2pi
2pi
2p i
809 2pi m
gðe m zÞ ¼ e m gðzÞ ) f ðzÞ ¼ hðe m zÞ þ gðe m zÞ ¼ e m hðzÞ þ e m gðzÞ ¼ e m hðzÞ þ e
gðzÞ )
1 sin mp f ðzÞ i cos mp
¼ ðhðzÞ gðzÞÞ In this step if we use Corollary 2.6, we obtain the desired result. h Corollary 2.8. If f ¼ hðzÞ þ gðzÞ be an element of S HðmÞ, then
u1 ¼ cos
2p 2p v2 u2 þ sin m m
2p 2p sin v2 ; u2 þ cos m m
v1 ¼
where hðzÞ ¼ u1 þ iv 1 ; gðzÞ ¼ u2 þ iv 2 Proof. Using the definition of S HðmÞ and Corollary 2.7 then we can write,
f ðzÞ ¼ hðzÞ þ gðzÞ i
sin mp f ðzÞ ¼ hðzÞ gðzÞ: cos mp
Therefore we have
hðzÞ ¼
1 p eim f ðzÞ 2 cos mp
gðzÞ ¼
1 p im p e f ðzÞ 2 cos m
These equalities shows that this corollary is true. Thus we can say that if f ðzÞ 2 S HðmÞ, then hðzÞ and gðzÞ are linear dependent. We consider, this property is invariant for symmetric harmonic mappings. It has not been proved, yet. h Corollary 2.9. If f ¼ hðzÞ þ gðzÞ be an element of S HðmÞ, since for m ¼ 2, the functions hðzÞ and gðzÞ becomes odd then we have, (i)
ð1 jb1 jÞ
Z
1
3
ð1 r 2 Þ3 dr 3
0
ð1 þ r 2 Þ2 ð1 þ jb1 jr 2 Þ
6 jf j 6 ð1 þ jb1 jÞ
Z
1
3
ð1 þ r 2 Þ2 dr 3
0
ð1 r 2 Þ2 ð1 þ jb1 jr 2 Þ
(ii)
1
jb1 j r2 ð1 r 2 Þ2 3
ð1 jb1 jr2 Þð1 þ r2 Þ2
0
6 jg ðzÞj 6
1
jb1 j þ r2 ð1 þ r 2 Þ2 3
ð1 þ jb1 jr2 Þð1 r 2 Þ2
(iii)
r jb 1 j r 2 r jb1 j þ r 2 6 j gðzÞ j 6 ð1 jb1 jr2 Þð1 þ r2 Þ ð1 þ jb1 jr 2 Þð1 r 2 Þ
Proof. This corollary is a simple consequence of Corollaries 2.4 and 2.6. h References [1] [2] [3] [4] [5]
P. Duren, Univalent Functions, Grundiehren der Mathematischen Wissenchafter, vol. 259, Springer-Verlag, New York, 1983. P. Duren, Harmonic Mappings in the Plane, Cambridge Tracts in Mathematics, vol. 156, Cambridge University Press, Cambridge UK, 2004. S. Fukui, K. Sakaguchi, An extension of a theorem of S. Ruscheweyh, Bull. Fac. Ed. Wakayama Univ. Nat. Sci. 29 (1980) 1–3. A.W. Goodman, Univalent Functions, vols. I and II, Mariner publishing Co., Tampa Fl, 1983. I. Graham, Gabriela Kohr, Geometric function theory in one and higher dimensions, Pure and Applied Mathematics, A Dekker series Monographs and Textbooks, (2003).