Convex Functions in a Half-plane - EMIS

Journal of Inequalities in Pure and Applied Mathematics http://jipam.vu.edu.au/ Volume 4, Issue 5, Article 102, 2003

CONVEX FUNCTIONS IN A HALF-PLANE NICOLAE N. PASCU AND NICOLAE R. PASCU “T RANSILVANIA “ U NIVERSITY OF B RA SOV ¸ S TR . I ULIU M ANIU N R . 50 B RA SOV ¸ – C OD 2200, ROMANIA . [email protected] G REEN M OUNTAIN C OLLEGE O NE C OLLEGE C IRCLE P OULTNEY, VT 05764, U.S.A. [email protected]

Received 24 March, 2003; accepted 16 August, 2003 Communicated by H.M. Srivastava

A BSTRACT. The class of convex hydrodynamically normalized functions in a half-plane was introduced by J. Stankiewicz. In this paper we introduce the general class of convex functions in the upper half-plane D (not necessarily hydrodynamically normalized) and we obtain necessary and sufficient conditions for an analytic function in D, to be convex univalent in D. Key words and phrases: Univalent function, Convex function, Half-plane. 2000 Mathematics Subject Classification. 30C45.

1. I NTRODUCTION We denote by D the upper half-plane {z ∈ C : Im (z) > 0}, by H the class of analytic functions in D, and by H1 the class of functions f ∈ H satisfying: (1.1)

lim [f (z) − z] = 0.

D3z→∞

The normalization (1.1) is known in the literature as hydrodynamic normalization, being related to fluid flows in Mechanics. The notion of convexity for functions belonging to the class H1 was introduced by J. Stankiewicz and Z. Stankiewicz ([4], [5]) as follows: Definition 1.1. The function f ∈ H1 is said to be convex if f is univalent in D and f (D) is a convex domain. ISSN (electronic): 1443-5756 c 2003 Victoria University. All rights reserved.

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We denote by CH1 (D) the class of convex functions satisfying the hydrodynamic normalization (1.1). J. Stankiewicz and Z. Stankiewicz obtained ([4], [5]) the following sufficient conditions for a function f ∈ H1 to be a convex function: Theorem 1.1. If the function f ∈ H1 satisfies: f 0 (z) 6= 0, for all z ∈ D and (1.2)

Im

f 00 (z) > 0, for all z ∈ D, f 0 (z)

then f is a convex function. The class of analytic univalent functions in a half-plane has been studied by F.G. Avhadiev [1] starting from the 1970’s. He examined the class of convex and univalent functions in a half plane that are not hydrodynamically normalized, obtaining the following theorem: Theorem 1.2. ([1]) The function f : D → C, analytic in D, is convex and univalent in D if and only if f 0 (i) 6= 0 and for any z ∈ D the following inequality holds:  
f 00 (z) 2 Im 2z + z + 1 0 > 0. f (z) Another result that characterizes the convexity property for univalent functions in the half plane that are not hydrodynamically normalized was obtained by the second author in [2]. After 1974, the year when Avhadiev’s paper was published, the only classes of univalent functions in the half-plane that had been studied were the univalent functions hydrodynamically normalized. We make the remark that the analytic representation of a geometric property (in this case the convexity property) is not unique. 2. M AIN R ESULTS The function ϕ : U → D given by 1−u 1+u is a conformal mapping of the unit disk U onto the upper half-plane D. For 0 < r < 1, the image of the disk Ur = {z ∈ C : |z| < r} under ϕ is the disk Dr = {z ∈ C : |z − zr | < Rr }, where:  1 + r2   ;  zr = i 1 − r2 (2.1) .    Rr = 2r 1 − r2 To see this, note that in polar coordinates u = reit , using the identity: 1 + re−it = 1 + reit ϕ (u) = i

we obtain:

1 − reit 1 + r2 2r (1 + re−it ) 2r i 1 + reit − i 1 − r2 = (1 + reit ) (1 − r2 ) = 1 − r2 , for any r ∈ (0, 1) and any t ∈ [0, 2π), which shows that the image under ϕ of the boundary of the disk Ur is the boundary of the disk Dr . Since ϕ (0) = i ∈ Dr , it follows that ϕ (Ur ) = Dr .

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Lemma 2.1. ([3])The family of domains {Dr }r∈(0,1) has the following properties: i) for any positive real numbers 0 < r < s < 1 we have Dr ⊂ Ds ; ii) for any complex number z ∈ D there exists rz ∈ (0, 1) such that z ∈ Dr , for any r ∈ (rz , 1); iii) for any z ∈ D and r ∈ (rz , 1) arbitrarily fixed, there exists ur ∈ U such that z = zr + Rr ur . Moreover, we have the following equalities:  limur = −i  r→1

.

 limRr (1 − |ur |) = Im z r→1

i) For any 0 < r < s < 1 we have:

Proof.

Dr = ϕ (Ur ) ⊂ ϕ (Us ) = Ds . ii) For z ∈ D we have ϕ−1 (z) ∈ U , hence considering rz = |ϕ−1 (z)| we have rz ∈ (0, 1), and for any r ∈ (rz , 1) we obtain z ∈ ϕ (Ur ) = Dr . iii) If z = X + iY is an arbitrarily fixed point in Dr , r ∈ (rz , 1), then the complex number ur = xr + iyr given by: z − zr ur = Rr has the property that |ur | < 1. Using the relations (2.1) we get:   2r 1 + r2 2r X + iY = xr + i + yr 1 − r2 1 − r2 1 − r2 and therefore:  2   xr = 1 − r X,   2r  2 2    yr = (1 − r ) Y − (1 + r ) , 2r hence it follows: (1 − r2 ) (1 − r2 ) Y − (1 + r2 ) lim ur = lim X +i r→1 r→1 2r 2r = −i, and 2

2

2r 4r2 − |z|2 (1 − r2 ) + 2(1 − r4 )Y − (1 + r2 ) · r→1 1 − r 2 4r2
1 − r2 |z|2 (1 − r2 ) = lim − + Y 1 + r2 − r→1 2r 2r = 2Y = 2 Im z.


lim Rr 1 − |ur |2 = lim

r→1

As limr→1 1 + |ur | = 2, follows from the previous inequality, the final result follows from the second part of iii), completing the proof.  The next theorem is obtained as a consequence of “the second coefficient inequality” for univalent functions in the unit disk, due to Bieberbach:

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Theorem 2.2. If g : U → C is analytic and univalent in U , then for any z ∈ U the following inequality holds:
zg 00 (z) 2 2 −2 |z| + 1 − |z| ≤ 4 |z| . g 0 (z) Using Lemma 2.1 we obtain the following result, which corresponds to the previous theorem in the case of univalent functions in the half-plane: Theorem 2.3. ([3]) If the function f : D → C is analytic and univalent in the half-plane D, then for any z ∈ D we have the inequality f 00 (z) (2.2) i − Im (z) f 0 (z) ≤ 2. The equality is satisfied for the function given by f (z) = z 2 at the point z = i. We make the observation that a simple function such as f : D → C defined by √ f (z) = z, (where we consider a fixed branch of the logarithm for the square root) is univalent in the domain D, f (D) is a convex domain, yet the function f is not considered to be convex in the sense of Definition 1.1 since it does not belong to the class H1 (f does not satisfy the hydrodynamic normalization (1.1)). This observation suggested the idea that it is necessary to give up the hydrodynamic normalization condition, a much too restrictive normalization. In this sense we propose a new definition of convexity for analytic functions in D, to include a larger class of analytic functions in D, not necessarily hydrodynamically normalized: Definition 2.1. A function f ∈ H is said to be convex in D if f is univalent in D and f (D) is a convex domain. We will denote by C(D) the class of convex functions (in the sense of Definition 2.1). The next theorem gives necessary and sufficient conditions for a function f ∈ H to belong to the class C(D): Theorem 2.4. For an analytic function f : D → C, the following are equivalent: i) f ∈ C(D); ii) f 0 (iy) 6= 0 for any y > 1, and for any r ∈ (0, 1) and z ∈ Dr the following inequality holds: (2.3)

(2.4)

Re

(z − zr ) f 00 (z) + 1 > 0, f 0 (z)

where Dr is the disk {z ∈ C : |z − zr | < Rr } and  1 + r2   z = i ,  r  1 − r2     Rr = 2r . 1 − r2

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Proof. Given the function f ∈ C(D), denote by ∆ the convex domain f (D). The function ϕ : U → D given by 1−u ϕ (u) = i 1+u represents conformally the disk U to the half-plane D, and for any r ∈ (0, 1) we have ϕ (Ur ) = Dr . The function f ◦ ϕ : U → C represents conformally the unit disk U onto ∆ = f (D). Since the domain ∆ is convex, it follows that the function f ◦ ϕ is convex and univalent in the unit disk U , and hence represents conformally any disk Ur (0 < r < 1), onto a convex domain. Since ϕ (Ur ) = Dr , it follows that for any r ∈ (0, 1) the domain ∆r = f (Dr ) is convex. For r ∈ (0, 1) arbitrarily fixed, the function gr : U → C given by gr (u) = f (zr + Rr u) ,

(2.5)

where zr , Rr are given by (2.4), represents conformally the disk U onto the convex domain ∆r . Using the results for convex and univalent functions in the unit disk, it follows that the domain ∆r is convex if and only if gr0 (0) = Rr f 0 (zr ) 6= 0

(2.6)

and for any u ∈ U the following inequality holds: (2.7)

Re

zgr00 (u) zRr f 00 (zr + Rr u) + 1 = Re + 1 > 0. gr0 (u) f 0 (zr + Rr u)

Denoting z = zr + Rr u, and observing that u ∈ U if and only if z ∈ Dr , the previous inequality can be written as (2.8)

(z − zr ) f 00 (z) Re + 1 > 0, f 0 (z)

for any z ∈ Dr , proving the necessity for condition (2.3). 2 Since zr = i 1+r , for r ∈ (0, 1) we have: 1−r2 1 + r2 >1 1 − r2 for any r ∈ (0, 1) and thus the condition (2.6) is equivalent to f 0 (iy) 6= 0 for any y > 1. Conversely, if ii) holds, then for any arbitrarily fixed r ∈ (0, 1) the function gr (u) = f (zr + Rr u) is convex and univalent in the disk U . It follows that for any r ∈ (0, 1) the domain ∆r = gr (U ) is convex, and since ∆r = f (Dr ), it follows S that the function f is convex and univalent in the domain Dr , for any r ∈ (0, 1). Since r∈(0,1) Dr = D, it follows that the function f is convex and univalent in the half-plane D, completing the proof.  |zr | =

In the previous proof we obtained the following result: Corollary 2.5. If the function f : D → C is convex and univalent in D, then ∆r = f (Dr ) is a convex domain for any r ∈ (0, 1). Remark 2.6. In [2] the second author introduced the subclass C1 (D) of the class of convex univalent functions as follows: Definition 2.2. ([2]) We say that the analytic function f : D → C belongs to the class C1 (D) if for any z ∈ D we have: (2.9)

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f 0 (z) 6= 0

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and

(2.10)

 zf 00 (z)   Re + 1 > 0,    f 0 (z)   f 00 (z)   > 0.  Im 0 f (z)

It is known that if the function g : U → C is convex and univalent in the unit disk U , with the Taylor series expansion: g (z) = z + a2 z 2 + · · · , then |a2 | ≤ 1. The class of convex and univalent functions in the unit disk, normalized by f (0) = f 0 (0) − 1 = 0 is denoted by C. The above property for functions belonging to the class C has the following important consequence: Theorem 2.7. If the function g : U → C belongs to the class C, then for any z ∈ U the following inequality holds:
zg 00 (z) 2 2 −2 |z| + 1 − |z| ≤ 2 |z| . g 0 (z) Using this result we obtain a differential characterization of the class C(D) of convex and univalent functions in the half-plane D: Theorem 2.8. If the function f : D → C belongs to the class C(D), then for any z ∈ D we have the inequality: f 00 (z) (2.11) i − Im (z) f 0 (z) ≤ 1. Proof. If the function f belongs to C(D), by Corollary 2.5 it follows that ∆r = f (Dr ) is a convex domain for any r ∈ (0, 1). The function gr given by formula (2.5) represents conformally the unit disk U onto gr (U ) = ∆r , and since ∆r is a convex domain, it follows that the function gr is convex and univalent. The function gr (u) − gr (0) f (zr + Rr u) − f (zr ) = 0 gr (0) Rr f 0 (zr ) is therefore convex and univalent in u ∈ U , normalized by gr (0) = gr0 (0) − 1 = 0 for any r ∈ (0, 1). By Theorem 2.7 it follows that for any r ∈ (0, 1) and any u ∈ U the following inequality holds:
uRr f 00 (zr + Rr u) 2 2 −2 |u| + 1 − |u| ≤ 2 |u| . (2.12) f 0 (zr + Rr u) Given z ∈ D, by Lemma 2.1 there exists rz ∈ (0, 1) such that for any fixed r ∈ (rz , 1), there is ur ∈ U such that z = zr + Rr ur ∈ Dr and    lim ur = −i, r→1

  lim (1 − |ur |) Rr = Im z. r→1

Considering u = ur in the inequality (2.12) and passing to the limit with r → 1, we obtain: 00 −2 + 2 Im(z) −if (z) ≤ 2. f 0 (z)

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Since z ∈ D was arbitrarily chosen, we have shown that for any z ∈ D the following inequality holds: 00 i − Im(z) f (z) ≤ 1, f 0 (z) 

and the theorem is proved. The next result is an important consequence of Theorem 2.8:

Corollary 2.9. If the function f : D → C is convex and univalent in the half-plane D, then for any z ∈ D we have the inequality: f 00 (z) (2.13) Im 0 > 0. f (z) Proof. If the function f is convex and univalent in the half-plane D, by the inequality (2.11) 00 (z) given by Theorem 2.8, it follows that for any z ∈ D, the point w = Im (z) ff 0 (z) belongs to the disk centered at i with radius 1. Since this disk belongs to the upper half-plane, it follows that for any z ∈ D the inequality (2.13) holds.  Remark 2.10. The result in the previous corollary was obtained, using different methods, by F.G. Avhadiev [1]. Example 2.1. The function f : D → C given by f (z) = z a , is convex and univalent for any a ∈ [−1, 0)∪(0, 1], since the function f is analytic and univalent in D, and the domains: f (D) = {z ∈ C : arg (z) ∈ (0, aπ)}, for a ∈ (0, 1), and f (D) = {z ∈ C : arg (z) ∈ (aπ, 0)}, for a ∈ (−1, 0), are convex. The following inequalities hold: z − zr (z − zr ) f 00 (z) Re + 1 = (a − 1) Re +1 0 f (z) z a |z 2 | − |zr | (a − 1) Im z = |z|2   a−1 a 2 2 = 2 (Re z) + (Im z) − |zr | Im z . a |z| Let us observe that if a ∈ [−1, 0), then for any r ∈ (0, 1), we have the following inequality: a−1 (Re z)2 + (Im z)2 − |zr | Im z < 0, a r| r| for any z in the disk centered at i(a−1)|z with radius (a−1)|z . Since 2a 2a a−1 |zr | ≥ |zr | , 2a this disk is contained in the disk Dr , and hence by Theorem 2.4 it follows that for a ∈ [−1, 0) we have f ∈ C(D). For a ∈ (0, 1] we have: Re

(z − zr ) f 00 (z) Im z + 1 = a − (a − 1) |zr | 2 > 0 0 f (z) |z |

for any r ∈ (0, 1) and for any z ∈ D, and therefore by Theorem 2.4 the function f belongs to the class C(D) for a ∈ (0, 1] as well.

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N ICOLAE N. PASCU AND N ICOLAE R. PASCU

Applying Theorem 1.2 to the same function f , we obtain:   (z 2 + 1) f 00 (z) (z 2 + 1) (a − 1) Im 2z + = 2y + Im f 0 (z) z  −2  = |z| y (a + 1) |z|2 − (a − 1) > 0 for any z ∈ D, if and only if a ∈ [−1, 1]. The condition f 0 (i) is satisfied for a 6= 0, hence it follows that f ∈ C(D) for any a ∈ [−1, 0) ∪ (0, 1]. Trying to apply the result due to J. Stankiewicz, we can see that f ∈ / CH1 (D) for any value of a ∈ [−1, 0) ∪ (0, 1] since the considered function f satisfies the hydrodynamic normalization just for a = 1, but in this case f 00 (z) Im 0 = 0, f (z) and the condition obtained by J. Stankiewicz is not satisfied. We therefore have the inclusion CH1 (D) ( C(D). R EFERENCES [1] F.G. AVHADIEV, O nekotorah odnolistnostâh otobrajeniah poluploskosti, Trudâ Seminara po kraevâm zadaciam, 11 (1974). [2] N.R. PASCU, On a class of convex functions in a half-plane, General Mathematics, 8(1-2) (2000). [3] N.N. PASCU, On univalent functions in a half-plane, Studia Univ.“Babes-Bolyai”, Math., XLVI(2) (2001). [4] J. STANKIEWICZ AND Z. STANKIEWICZ, On the classes of functions regular in a half-plane I, Bull. Pollisch Acad. Sci. Math., 39(1-2) (1991), 49–56. [5] J. STANKIEWICZ, Geometric properties of functions regular in a half-plane, Current Topics in Analytic Function Theory, World Sci. Publishing, River Edge NJ (1992), pp. 349–362.

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