U N I V E R S I T A T I S M A R I A E C U R I E - S K Ł O D O W S K A L U B L I N – P O L O N I A
VOL. LXIV, NO. 1, 2010 SECTIO A 81–91
AGNIESZKA SIBELSKA
On the order of starlikeness and convexity of complex harmonic functions with a two-parameter coefficient condition
Abstract. The article of J. Clunie and T. Sheil-Small [3], published in 1984, intensified the investigations of complex functions harmonic in the unit disc Δ. In particular, many papers about some classes of complex mappings with the coefficient conditions have been published. Consideration of this type was undertaken in the period 1998–2004 by Y. Avci and E. Złotkiewicz [2], A. Ganczar [5], Z. J. Jakubowski, G. Adamczyk, A. Łazińska and A. Sibelska [1], [8], [7], H. Silverman [12] and J. M. Jahangiri [6], among others. This work continues the investigations described in [7]. Our results relate primarily to the order of starlikeness and convexity of functions of the aforementioned classes.
1. Introduction. Let
Δ := {z ∈ C : |z| < 1},
Δr := {z ∈ C : |z| < r}, r > 0, A :={(α, p) ∈ R2 : 0 ≤ α ≤ 1, p > 0},
Un(α, p) := α np+ (1 − α)np+1, n = 2, 3, . . . , (α, p) ∈ A.
In [7] the following classes of complex harmonic functions have been in- troduced:
2000 Mathematics Subject Classification. 30C45, 31A05.
Key words and phrases. Complex harmonic functions, analytic conditions, convexity of orderβ, starlikeness of order β.
Definition 1. Let (α, p) be a fixed pair of parameters in the set A. By HS(α, p) we denote the class of the functions f of the form:
(1.1)
f (z) = h(z) + g(z), h(z) = z +
∞ n=2
anzn, g(z) =
∞ n=1
bnzn, z∈ Δ, |b1| < 1, satisfying the condition
(1.2) |b1| +∞
n=2
Un(α, p)
|an| + |bn|
≤ 1.
Definition 2. Let (α, p) ∈ A be fixed. Let us denote HS0(α, p) := {f ∈ HS(α, p) : b1= 0}.
The classes HS(1, 1), HS0(1, 1), HS(1, 2), HS0(1, 2) were investigated in the paper [2]. The results contained in [5] refer to the classes HS(1, p), HS0(1, p), p > 0 and in [8] the classes HS(α, 1), HS0(α, 1) for α ∈ [0, 1]
were considered.
In this paper we will show results related to the order of starlikeness and convexity of functions which belong to the aforementioned classes.
Recall the definition of complex harmonic functions starlike (convex) and starlike (convex) of the order β, β∈ [0, 1).
Definition 3. A univalent and sense-preserving complex harmonic function f of the form (1.1) is called starlike with respect to the origin (starlike) in Δ, if f(Δ) is a domain starlike with respect to the origin.
Definition 4. A univalent and sense-preserving complex harmonic function f of the form (1.1) is called convex in Δ if f (Δ) is a convex domain.
Remark 1. It is known that in order to prove starlikeness of the image of the disc Δ by a univalent sense-preserving mapping f it is sufficient to prove starlikeness of f (Δr) for every r ∈ (0, 1), i.e. to show that for any r ∈ (0, 1) we have
∂
∂θ
arg f(reiθ)
> 0, θ∈ [0, 2π].
By analogy, in order to prove convexity of the image of the disc Δ by a univalent sense-preserving mapping f , it is sufficient to prove convexity of f (Δr) for every r ∈ (0, 1), i.e. to show that for any r ∈ (0, 1) we have
∂
∂θ
arg ∂
∂θ(f(reiθ))
> 0, θ∈ [0, 2π].
Definition 5. Let β ∈ [0, 1). A univalent and sense-preserving complex harmonic function f of the form (1.1) is called starlike of the order β with respect to the origin in Δ if for any r∈ (0, 1), we have
(1.3) ∂
∂θ
arg f(reiθ)
> β, θ∈ [0, 2π].
Definition 6. Let β ∈ [0, 1). A univalent and sense-preserving complex harmonic function f of the form (1.1) is called convex of the order β in Δ if for any r∈ (0, 1), we have
(1.4) ∂
∂θ
arg ∂
∂θ(f(reiθ))
> β, θ∈ [0, 2π].
Remark 2. Definitions 5 and 6 are analogues of the appropriate classical definitions for the normalized holomorphic functions in Δ. In [9], among other things, the definition of a starlike (convex) function f of the form f (z) = z + a2z2+ . . . , z ∈ Δ is stated. The subclass of the class of starlike (convex) functions satisfying the condition Re{zf(z)/f(z)} > β (Re{1 + zf(z)/f(z)} > β), z ∈ Δ, 0 < β < 1 is called the class of functions starlike (convex) of the order β.
In his paper B. Pinchuk ([10]) noted that in 1936 M. S. Robertson ([11]) introduced more restrictive definitions of the classes of holomorphic func- tions starlike (convex) of the order β. Namely, holomorphic, normalized starlike (convex) functions were called by him starlike (convex) functions of the order β, β ∈ [0, 1), if they satisfy the condition Re{zf(z)/f(z)} ≥ β, z ∈ Δ (Re{1 + zf(z)/f(z)} ≥ β, z ∈ Δ) and for any sufficiently small
> 0 there exists a point z0 ∈ Δ such that Re{z0f(z0)/f(z0)} < β + (Re{1 + z0f(z0)/f(z0)} < β + ). The second part of Robertson’s defini- tion is overlooked by his followers.
In this paper we use generalization of the classical definition of the func- tions starlike (convex) of the order β, for complex harmonic functions.
It is worth remembering that the property of starlikeness (convexity) is hereditary for functions holomorphic in Δ, but for complex harmonic functions it need not be so [4].
Let α∈ [0, 1] and
p1(α) := 1 − log2(2 − α), p2(α) := 1 + p1(α),
p3(α) := 1 + p2(α), log21 = 0.
Let us denote
A1 := {(α, p) ∈ A : p ≥ p1(α)}, A2 := {(α, p) ∈ A : p ≥ p2(α)}, A3 := {(α, p) ∈ A : p ≥ p3(α)}.
α p
1 2 3
1 A1
A2
A3
p = p1(α) p = p2(α) p = p3(α)
A3⊂ A2⊂ A1
In [7] the following theorems are proved:
Theorem 1. Let (α, p) ∈ A1. Then HS(α, p) is the class of univalent and sense-preserving functions in Δ.
A1 is the largest set, in which every function f ∈ HS(α, p), (α, p) ∈ A is univalent in Δ.
Theorem 2. If (α, p) ∈ A1, then the functions of the class HS0(α, p) are starlike in Δ. If (α, p)∈ A2, then the functions of the class HS0(α, p) are convex in Δ.
Remark 3. The sets A1and A2 are the largest subsets of A, in which every function f ∈ HS0(α, p) is starlike, convex in Δ, respectively.
2. Main results. In view of Theorem 2 it seems natural to ask a question about the order of starlikeness of functions of the class HS0(α, p), (α, p) ∈ A1 and about the order of convexity of functions of the class HS0(α, p), (α, p) ∈ A2. The next theorems solve this problem.
Theorem 3. Let (α, p) ∈ A1. If f ∈ HS0(α, p), then f is a univalent sense-preserving function starlike of the order β∗(α, p) := UU2(α,p)−2
2(α,p)+1 in Δ.
Proof. The univalence and sense-preservation of the function f in Δ are guaranteed by Theorem 1.
Using a method similar to previously used in [2], J. M. Jahangiri ([6]) has proved that if a function f of the form (1.1) satisfies the condition
∞ n=1
n− β
1 − β|an| + n + β 1 − β|bn|
≤ 2, 0 ≤ β < 1,
then for every r∈ (0, 1), θ ∈ [0, 1], the condition (1.3) holds, so f is starlike of the order β in Δ.
If f belongs toHS0(α, p), the aforementioned coefficient condition takes the form
(2.1)
∞ n=2
n− β
1 − β|an| + n + β 1 − β|bn|
≤ 1, 0 ≤ β < 1.
It is obvious that for any β ∈ [0, 1) we have n−β1−β ≤ n+β1−β, n = 2, 3, . . . . Therefore, it is sufficient to show that β∗ is the largest constant such that for every fixed (α, p) ∈ A1, we have β∗ ∈ [0, 1) and, due to the condition (1.2), n+β1−β∗∗ ≤ Un(α, p) for any n = 2, 3, . . . .
We have 0 ≤ β∗(α, p) < 1 if (α, p) ∈ A1. Indeed, the inequality p ≥ p1(α), α ∈ [0, 1] is equivalent to 2p(2 − α) ≥ 2, i.e. U2(α, p) ≥ 2, therefore β∗(α, p) ≥ 0, (α, p) ∈ A1. The upper estimation is immediate.
The inequality n+β1−β∗∗ ≤ Un(α, p), n = 2, 3, . . . is equivalent to the inequal- ity
(2.2) β∗(α, p) ≤ Un(α, p) − n
Un(α, p) + 1, n = 2, 3, . . . , (α, p) ∈ A1. Let us consider the function of the form
t(x) = t(α, p; x) := xp(α + (1 − α)x) − x xp(α + (1 − α)x) + 1,
x≥ 2, (α, p) ∈ A1. We will prove that minx≥2t(x) = t(2). We have t(x) = xp+1p(1− α) + xp(1 − 2α + p) + xp−1pα− 1
[xp(α + (1 − α)x) + 1]2 , x≥ 2.
Let us denote s(x) := xp+1p(1− α) + xp(1 − 2α + p) + xp−1pα− 1, x ≥ 2, (α, p) ∈ A1. Then we have s(x) = pxp−2
(p + 1)(1 − α)x2+ (1 − 2α + p)x + α(p− 1)
, x≥ 2.
Let us consider three cases:
1) Let α ∈ [0, 1] and p ≥ 1. Then s(x) > 0, x ≥ 2, therefore s is an increasing function in (2, +∞). Moreover, s(2) ≥ 0.
Indeed, s(2) = 2p−1(2 − α)(3p + 4) − 6 · 2p−1− 1 ≥ 2p−1(3p − 2) − 1. Let us note that 3p− 2 ≥ 21−p for p≥ 1 and 2p−1(3p − 2) ≥ 1. Hence s(x) ≥ 0,
x ≥ 2, therefore t(x) ≥ 0, x ≥ 2 and consequently minx≥2t(x) = t(2) = β∗(α, p), α ∈ [0, 1], p ≥ 1.
2) Let now α ∈ (0, 1) and p1(α) ≤ p < 1. We will investigate the quadratic equation (p + 1)(1− α)x2+ (1 − 2α + p)x + α(p − 1), x ≥ 2 with zeros x1 = −1, x2 = (1−α)(p+1)α(1−p) ≤ 2. The last inequality is equivalent to p≥ 3α−22−α and χ(α) := 3α−22−α − p1(α) < 0, α ∈ (0, 1). Hence s(x) ≥ 0, x ≥ 2.
In this case we need to show that s(2)≥ 0, α ∈ (0, 1), p1(α) ≤ p < 1. We have
sp(2) = 2p−1
(2 − α)[(3p + 4) ln 2 + 3] − 6 ln 2 .
The inequality (2− α)[(3p + 4) ln 2 + 3] − 6 ln 2 ≥ 0 is equivalent to p ≥
2−α2 −3+4 ln 23 ln 2 .
Let us denote ψ(α) := 2−α2 − 3+4 ln 23 ln 2 , α ∈ (0, 1). We have ψ(α) < 0 for α ∈ (0, 1), so ψ(α) < p1(α) ≤ p, α ∈ (0, 1), i.e. s is an increasing function of variable p in (p1(α), 1), α ∈ (0, 1). Moreover, s(2)|p=p1(α) = 6 − 3 log2(2 − α) −(2−α)6 ≥ 0, α ∈ (0, 1).
Indeed, the last inequality has an equivalent form log2(2 − α) ≤ 2(1−α)2−α . Let us denote m(α) := log2(2 − α) − 2(1−α)2−α , α ∈ (0, 1). We have m(0) = m(1) = 0 and m(α) = 2(ln 2−1)+α
(2−α)2ln 2 . Hence m is a continuous function decreasing in (0, 2(1− ln 2)) and increasing in (2(1 − ln 2), 1). Therefore m(α) < 0, α∈ (0, 1).
The above considerations imply s(x) ≥ 0, x ≥ 2, therefore t(x) ≥ 0, x≥ 2, hence minx≥2t(x) = t(2) = β∗(α, p), α ∈ (0, 1), p1(α) ≤ p < 1.
3) Finally let us consider the case when α = 0, p ∈ (0, 1). Then s(x) = p(p + 1)xp−2(x2 + x) > 0, x ≥ 2 and s(2) = 2p(3p + 1) − 1 > 0, hence minx≥2t(x) = t(2) = β∗(0, p), p ∈ (0, 1).
Therefore, we showed that for any point (α, p)∈ A1there is minx≥2t(x) = t(2). From the form of the function t we conclude that β∗(α, p), (α, p) ∈ A1, is the largest constant such that for every fixed (α, p) ∈ A1 the system of conditions (2.2) holds for any n = 2, 3, . . . .
Thus, for each function of the class HS0(α, p), (α, p) ∈ A1, with any β ∈ [0, β∗(α, p)], the coefficient condition (2.1) holds. Hence the proof is
completed.
Proposition 1. In every class HS0(α, p), (α, p) ∈ A1 there exists a func- tion starlike of the order β∗(α, p) in the sense of the restrictive Robertson’s definition ([11]). The following formula
(2.3) f2∗(α, p; z) = z + b∗2z2, z∈ Δ, (α, p) ∈ A1, b∗2= 1 U2(α, p). gives an example of such function.
Proof. Since
|b∗1| +∞
n=2
Un(α, p)(|a∗n| + |b∗n|) = U2(α, p)|b∗2| = 1,
we have f2∗ ∈ HS0(α, p), (α, p) ∈ A1. Therefore, according to Theorem 3, the function f2∗ is starlike of the order β∗(α, p) = 1−2b1+b∗∗2
2 , β∗(α, p) ∈ [0, 1).
Therefore, for any r∈ (0, 1), θ ∈ [0, 2π] we have
∂
∂θ
arg f2∗(reiθ)
= Re
1 − 2b∗2e−3iθr 1 + b∗2e−3iθr
≥ β∗(α, p).
Putting θ = 0, we obtain 1 − 2b∗2r
1 + b∗2r ≥ β∗(α, p), r∈ (0, 1).
Let us denote u(r) := 1−2b1+b∗∗2r
2r, r ∈ [0, 1). The function u is continuous and decreasing in (0, 1), so for any r ∈ [0, 1), u(r) > 1−2b1+b∗∗2
2 . Moreover, limr→1−u(r) = 1−2b1+b∗∗2
2 . Hence the function f2∗ cannot be starlike of the order higher than β∗(α, p).
Let us note that for the function f2∗(α, p; ·) the equality in condition (2.1)
holds for β = β∗(α, p).
Property 1. Let (α, p) ∈ A1, a ≥ 0. If (α, p) is any point on the curve p = pa(α), α ∈ [0, 1], where pa(α) := p1(α) + a, then HS0(α, p) is the class of functions starlike of the order β∗(a) := β∗(α, pa(α)) = 221+a1+a−2+1 and in this class there exists a function starlike of the order β∗(a), in the sense of the restrictive Robertson’s definition ([11]).
If a = 0 (p0(α) = p1(α)), then β∗(0) = β∗(α, p1(α)) = 0.
Moreover, if 0≤ a1 < a2, then 0≤ β∗(a1) < β∗(a2) < 1.
Indeed, let us note that
U2(α, pa(α)) = α21+a−log2(2−α)+ (1 − α)22+a−log2(2−α) = 21+a, hence β∗(α, pa(α)) = 221+a1+a−2+1. Moreover, U2(α, p0(α)) = 2, α ∈ [0, 1], so if (α, p) is a point on the curve p = p1(α), α ∈ [0, 1], which is the arc of boundary of the set A1, then functions of the classHS0(α, p) are starlike of the order 0, so starlike in Δ.
Moreover, (β∗(a)) = (221+a1+a3 ln 2+1)2 > 0, a ≥ 0, therefore, the function β∗ is an increasing mapping of variable a. We also have the following result.
Theorem 4. Let (α, p) ∈ A2. If f ∈ HS0(α, p), then f is an univalent sense-preserving function convex of the order βc(α, p) := UU2(α,p)−4
2(α,p)+2 in Δ.
Proof. Univalence and sense-preservation of the function f in Δ are guar- anteed by Theorem 1.
We will use the fact (see [6]) that if f of the form (1.1) (b1= 0) satisfies the condition
(2.4)
∞ n=2
n(n− β)
1 − β |an| + n(n + β) 1 − β |bn|
≤ 1, 0 ≤ β < 1,
then for any r ∈ (0, 1), f satisfies the condition (1.4), so it is a function convex of the order β in the disc Δ.
Using the fact that for any β ∈ [0, 1), n−β1−β ≤ n+β1−β, n = 2, 3, . . . , we will show that βc(α, p), (α, p) ∈ A2 is the largest constant from the range [0, 1) such that n(n+β1−βcc(α,p)(α,p)) ≤ Un(α, p), (α, p) ∈ A2 for any n = 2, 3, . . . .
It is obvious that βc(α, p) ∈ [0, 1) if (α, p) ∈ A2. Indeed, the inequality p≥ p2(α), α ∈ [0, 1] is equivalent to U2(α, p) ≥ 4.
Let us consider a function of the form a(x) = a(α, p; x) := xp(α+(1−α)x)−x2 xp(α+(1−α)x)+x, x≥ 2, (α, p) ∈ A2 and q := p− 1. Investigation of the behavior of the func- tion a(α, p;·), (α, p) ∈ A2 in the range [2, +∞) is equivalent to examination of a function t(α, q;·), (α, q) ∈ A1 considered in the proof of Theorem 3.
Therefore, we have minx≥2a(x) = a(2) = UU2(α,p)−4
2(α,p)+2, (α, p)∈ A2. Hence
∞ n=2
n(n− βc(α, p))
1 − βc(α, p) |an| +n(n + βc(α, p)) 1 − βc(α, p) |bn|
≤∞
n=2
n(n + βc(α, p))
1 − βc(α, p) (|an| + |bn|)
≤∞
n=2
Un(α, p)(|an| + |bn|) ≤ 1, (α, p) ∈ A2, so, if β = βc(α, p), then the condition (2.4) holds.
Therefore, for any function of the class HS0(α, p), (α, p) ∈ A2 and for each β ∈ [0, βc(α, p)], the coefficient condition (2.4) holds. It is a sufficient condition for convexity of the order β of a function f in Δ. Proposition 2. In each class HS0(α, p), (α, p) ∈ A2 there exists a function which is convex of the order β∗(α, p) in the sense of the restrictive Robert- son’s definition ([11]). Formula (2.3) gives an example of such function.
In the proof we use the same method, as in the proof of Proposition 1.
Property 2. Let (α, p) ∈ A2, a ≥ 0. If (α, p) is any point of the curve p = pb(α), α ∈ [0, 1], where pb(α) := p2(α) + b, then HS0(α, p) is the class of the functions convex of the order βc(b) := βc(α, pb(α)) = 222+b2+b−4+2 and in
this class there exists a function convex of the order βc(b) in the sense of the restrictive Robertson’s definition ([11]).
Moreover, if 0≤ b1< b2, then 0≤ βc(b1) < βc(b2) < 1.
Justification of validity of this property is analogous to the proof of Prop- erty 1.
3. On other properties of functions of the classes HS(α, p) and HS0(α, p). The very well-known Alexander theorem for univalent holo- morphic functions shows relationships between starlike and convex func- tions. P. Duren ([4], p. 108) gave the partial extension of this theorem in the case of complex harmonic functions. We show an analogous extension for functions of the classes investigated in this paper.
Property 3. Let α ∈ [0, 1], p > 1. If f of the form (1.1) is a function of the classHS(α, p − 1) (HS0(α, p − 1)), then a function K of the form
K(z) =
1 0
f (zt) t dt =
z
0
h(u) u du +
z
0
g(u)
u du, z∈ Δ belongs to the class HS(α, p) (HS0(α, p)).
Indeed, we have K(z) =
1 0
f (zt) t dt =
1 0
tz +∞
n=2antnzn
t + b1zt +∞
n=2bntnzn t
dt
=
1 0
z + b1z +
∞ n=2
anzntn−1+∞
n=2
bnzntn−1
dt
= z +
∞ n=2
anzn
n + b1z +
∞ n=2
bnzn
n , z∈ Δ, therefore, K is of the form (1.1).
Moreover,
|b1| +∞
n=2
Un(α, p)|an| n + |bn|
n
= |b1| +∞
n=2
Un(α, p − 1)(|an| + |bn|) ≤ 1.
The next theorems concern the convolutions of complex harmonic func- tions in Δ.
Definition 7. Let fk(z) = hk(z) + gk(z), where hk(z) = z +∞
n=2a(k)n zn, gk(z) =∞
n=1b(k)n zn are holomorphic in Δ, k = 1, 2.
Hadamard’s convolution of the functions f1and f2is given by the formula (f1∗ f2)(z) := z +∞
n=2
a(1)n a(2)n zn+∞
n=1
b(1)n b(2)n zn, z∈ Δ.
Definition 8. Let fk(z) = hk(z) + gk(z), where hk(z) = z +∞
n=2a(k)n zn, gk(z) =∞
n=1b(k)n zn are holomorphic in Δ, k = 1, 2.
The integral convolution of the functions f1and f2 is given by the formula (f1 f2)(z) := z +∞
n=2
a(1)n a(2)n
n zn+∞
n=1
b(1)n b(2)n
n zn, z∈ Δ.
We have the following result.
Theorem 5. Let ˜f (z) = z +∞
n=2˜anzn+∞
n=2˜bnzn, z∈ Δ be a univalent complex harmonic function convex in Δ.
If f ∈ HS0(α, p), (α, p) ∈ A2, then f∗ ˜f is a univalent sense-preserving function starlike in Δ and f ˜f is a univalent sense-preserving function convex in Δ.
If f ∈ HS0(α, p), (α, p) ∈ A3, then f∗ ˜f is a univalent sense preserving function convex in Δ.
Proof. It is obvious that in each considered case f ∗ ˜f and f ˜f are complex harmonic functions in Δ and have the form required in the classHS0(α, p).
Let us assume that (α, p)∈ A2. According to the fact that the estimations
|˜an| ≤ n+12 , |˜bn| ≤ n−12 , n = 2, 3, . . . hold ([3], th. 5.10), we have |˜ann| ≤ 1,
|˜bn|
n ≤ 1, n = 2, 3, . . . . Hence
∞ n=2
n(|an˜an| + |bn˜bn|) =
∞ n=2
n2
|an|
˜an n
+ |bn|
˜bn n
≤∞
n=2
n2(|an| + |bn|) ≤∞
n=2
Un(α, p)(|an| + |bn|) ≤ 1, so f∗ ˜f ∈ HS0(1, 1), therefore the convolution is a univalent sense-preserving starlike function.
Due to the fact that (f ˜f )(z) = 1
0 (f∗ ˜f)(zt)
t dt, z ∈ Δ and f ∗ ˜f ∈ HS0(1, 1), we obtain from Property 3 that a function f ˜f belongs to the classHS0(1, 2), so it is a univalent and sense-preserving function convex in Δ (see [2]).
Let (α, p)∈ A3. Then it can be shown that Un(α, p) ≥ n3, n = 2, 3, . . . . Hence, using an analogous method as above, we can show that for the function f∗ ˜f the inequalities
∞ n=2
n2(|an˜an| + |bn˜bn|) ≤∞
n=2
n3(|an| + |bn|) ≤∞
n=2
Un(α, p)(|an| + |bn|) ≤ 1 hold. Therefore, for the convolution f∗ ˜f the condition sufficient for univa-
lence sense-preservation and convexity holds.
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Agnieszka Sibelska
Departament of Nonlinear Analysis
Faculty of Mathematics and Computer Science University of Łódź
ul. S. Banacha 22 90-238 Łódź Poland
e-mail: sibelska@math.uni.lodz.pl Received September 21, 2009
