New Properties for Starlike and Convex Functions of

Int. Journal of Math. Analysis, Vol. 4, 2010, no. 1, 39 - 62

New Properties for Starlike and Convex Functions of Complex Order Toshio Hayami and Shigeyoshi Owa Department of Mathematics, Kinki University Higashi-Osaka, Osaka 577-8502, Japan ha ya [email protected], [email protected] Abstract For functions f (z) which are starlike of complex order b (b = 0) in the open unit disk U introduced by M. A. Nasr and M. K. Aouf (J. Natural Sci. Math. 25(1985), 1 - 12), in consideration of its properties, some sufficient conditions, necessary conditions and distortion theorems with coefficient inequalities of f (z) as the improvement of the well-known result due to H. Silverman (Proc. Amer. Math. Soc. 51(1975), 109 - 116) are discussed. Furthermore, some new coefficient inequalities including some new interesting corollaries are discussed.

Mathematics Subject Classification: 30C45 Keywords: Coefficient inequality, analytic function, univalent function, starlike function of complex order, λ-spiral like function

1

Introduction and Preliminaries Let A be the class of functions f (z) of the form

(1.1)

f (z) = z +

∞

an z n

(a0 = 0, a1 = 1)

n=2

which are analytic in the open unit disk U = {z ∈ C : |z| < 1}. Furthermore, let P denote the class of functions p(z) of the form (1.2)

p(z) = 1 +

∞ n=1

pn z n

40

T. Hayami and S. Owa

which are analytic in U. If p(z) ∈ P satisfies Re p(z) > 0 (z ∈ U), then we say that p(z) is the Carathéodory function (cf. [1]). If f (z) ∈ A satisfies the following inequality zf (z) Re >α (z ∈ U) f (z) for some α (0 α < 1), then f (z) is said to be starlike of order α in U. We denote by S ∗ (α) the subclass of A consisting of functions f (z) which are starlike of order α in U. Similarly, we say that f (z) is a member of the class K(α) of convex functions of order α in U if f (z) ∈ A satisfies the following inequality zf (z) Re 1 + >α (z ∈ U) f (z) for some α (0 α < 1). As usual, in the present investigation, we write S ∗ = S ∗ (0) and K = K(0). Classes S ∗ (α) and K(α) were introduced by Robertson [5]. Next, a function f (z) ∈ A is called λ-spiral like of order α in U if and only if zf (z) iλ −α >0 (z ∈ U) Re e f (z) for some real λ (− π2 < λ < SP(λ, α).

π ) 2

and α (0 α < 1). We denote this class by

Moreover, for some non-zero complex number b, we consider the subclasses Sb∗ and Kb of A as follows: 1 zf (z) ∗ −1 > 0 (z ∈ U) Sb = f (z) ∈ A : Re 1 + b f (z) and

1 zf (z) Kb = f (z) ∈ A : Re 1 + > 0 (z ∈ U) . b f (z)

If a function f (z) belongs to the class Sb∗ or Kb , we say that f (z) is starlike or convex of complex order b (b = 0), respectively. In [3], Nasr and Aouf introduced the class Sb∗ .

41

Starlike and convex functions of complex order

Then, we can see that ∗ S1−α = S ∗ (α),

∗ K1−α = K(α) and S(1−α)e −iλ cos λ = SP(λ, α).

Example 1.1 n

j + 2(b − 1)

(1.3)

fb (z) =

and (1.4)

gb (z) =

∞ z j=2 = z + 2b (1 − z) (n − 1)! n=2

z n ∈ Sb∗

⎧ n

⎪ ⎪ j + 2(b − 1) ⎪ ∞ ⎪ 1−2b ⎪ 1 − (1 − z) j=2 ⎪ ⎪ =z+ z n ∈ Kb ⎪ ⎨ 1 − 2b n! n=2

⎪ ⎪ ⎪ ∞ ⎪ 1 ⎪ 1 1 n ⎪ ⎪ 1 = K z ∈ K . log = z + ⎪ 2 ⎩ 2 1−z n n=2

Theorem 1.2 For a function f (z) ∈ A, it follows that f (z) ∈ Kb and f (z) ∈ S

Proof.

∗

1

z{f (z)} b ∈ S ∗

if and only if

z

if and only if 0

f (ξ) ξ

b dξ ∈ Kb .

1

We set F (z) = z{f (z)} b . If f (z) ∈ Kb , then zF (z) 1 zf (z) Re = Re 1 + > 0. F (z) b f (z) 1

Therefore, F (z) = z{f (z)} b ∈ S ∗ . The converse is also proved. Setting b = 1 − α in Theorem 1.2, we have Corollary 1.3 For a function f (z) ∈ A, it follows that f (z) ∈ K(α) if and only if

1

z{f (z)} 1−α ∈ S ∗

(b = 0)

1 b = 2

42


and f (z) ∈ S

∗

z

if and only if 0

f (ξ) ξ

1−α dξ ∈ K(α).

Silverman [6] has proved the following well-known theorem. Theorem 1.4 If a function f (z) ∈ A satisfies the following inequality ∞

(n − α)|an | 1 − α

n=2

or

∞

n(n − α)|an | 1 − α

n=2

for some α (0 α < 1), then f (z) ∈ S ∗ (α) or f (z) ∈ K(α), respectively. We apply the following lemma (see, [7]) to obtain our results. Lemma 1.5 A function p(z) ∈ P satisfies Re p(z) > 0 (z ∈ U) if and only if ζ −1 (z ∈ U) p(z) = ζ +1 for all |ζ| = 1.

Then, by using Lemma 1.5, various conditions for starlike functions are studied. The following results are enumerated as the some examples. Silverman, Silvia and Telage [7] have given Theorem 1.6 A function f (z) ∈ A is in S ∗ (α) if and only if ⎛

ζ + 2α − 1 2 ⎞ z z+ 1⎜ ⎟ 2 − 2α ⎝f (z) ∗ ⎠ = 0 2 z (1 − z)

(z ∈ U, |ζ| = 1)

where ∗ means the convolution or Hadamard product of two functions. Furthermore, letting α = 0 in Theorem 1.6, Nezhmetdinov and Ponnusamy [4] have given the sufficient conditions for coefficients of f (z) to be in the class S ∗.


43

Hayami, Owa and Srivastava [2] have shown the following results. Theorem 1.7 If f (z) ∈ A satisfies the following condition n k ∞ γ β (j + 1 − 2α)(−1)k−j aj n−k k−j n=2

j=1

k=1

k n γ β (j − 1)(−1)k−j aj + 2(1 − α) n−k k−j k=1

j=1

for some α (0 α < 1), β ∈ R and γ ∈ R, then f (z) ∈ S ∗ (α).

Theorem 1.8 If f (z) ∈ A satisfies the following condition n k ∞ γ β j(j + 1 − 2α)(−1)k−j aj n−k k−j n=2

j=1

k=1

k n γ β + j(j − 1)(−1)k−j aj 2(1 − α) n−k k−j k=1

j=1

for some α (0 α < 1), β ∈ R and γ ∈ R, then f (z) ∈ K(α).

Theorem 1.9 If f (z) ∈ A satisfies the following condition n k ∞ γ β k−j (j − α + (1 − α)e−2iλ )(−1) aj n−k k−j n=2

k=1

j=1

k n γ β k−j aj (j − 1)(−1) + 2(1 − α) cos λ n−k k−j k=1

j=1

for some α (0 α < 1), λ − π2 < λ < π2 , β ∈ R and γ ∈ R, then f (z) ∈ SP(λ, α).

44


Properties of the class Sb∗

2

Generally, for some b = 1 − α or b = (1 − α)e−iλ cos λ (0 α < 1, − π2 < λ < π2 ), it is well known that f (z) ∈ Sb∗ = S ∗ (α) or SP(λ, α) is univalent in U, but the almost f (z) ∈ Sb∗ is not univalent in U. Then, can we find the radius r (0 < r 1) for f (z) ∈ Sb∗ to be univalent in Ur = {z ∈ C : |z| < r}? One of the answer of this problem about Koebe type function fb (z) ∈ Sb∗ given by (1.3) is below. Theorem 2.1 fb (z) = where

r=

z ∈ Sb∗ belongs to the class S ∗ in Ur , (1 − z)2b

⎧ |1 − b| − |b| ⎪ ⎪ ⎪ ⎪ ⎨ 1 − 2Re(b)

⎪ ⎪ ⎪ ⎪ ⎩

1 2|b|

1 Re(b) = 2 1 Re(b) = 2

and fb (z) is also univalent in Ur . Proof.

Let us define w=

Then,

zfb (z) 1 + (2b − 1)z = . fb (z) 1−z

w−1 1)

and fb (z) is also univalent in Ur . zf (z) = u + iv and b = ρeiϕ (ρ > 0, 0 ϕ < 2π). Then, f (z) the condition of the definition of Sb∗ is equivalent to 1 zf (z) cos ϕ sin ϕ (2.1) Re 1 + −1 =1+ (u − 1) + v > 0. b f (z) ρ ρ Next, let F (z) =

Hence, u > tan(−ϕ)v + 1 −

ρ cos ϕ

(cos ϕ > 0)

u < tan(−ϕ)v + 1 −

ρ cos ϕ

(cos ϕ < 0).

or

We denote by d(l1 , 1) the distance between the boundary line l1 : (cos ϕ)u + (sin ϕ)v + ρ − cos ϕ = 0 of the half plane satisfying the condition (2.1) and the point F (0) = 1. A simple computation gives us that d(l1 , 1) =

| cos ϕ × 1 + sin ϕ × 0 + ρ − cos ϕ| = ρ, cos2 ϕ + sin2 ϕ

that is, that d(l1 , 1) is always equal to |b| = ρ regardless of ϕ. Thus, if we consider the circle C with center at 1 and radius ρ, then we can know the definition of Sb∗ means that F (U) is covered by the half plane separated by a tangent line of C and containing C. By this fact, we consider the perpendicular l2 : v = tan ϕ(u − 1) of the line l1 through the point 1, and also the intersection point z0 of l1 and l2 . And, we take the point ν on l2 , satisfying the following relation (2.2)

|1 − z0 | |ν − z0 | and

arg(1 − z0 ) = arg(ν − z0 ).

46


Namely, ν is farther from the point z0 than the point 1 and in the side same as the point 1. Then, we see that if F (U) is in the interior of the circle Cν with center at ν and radius |ν − z0 |, then f (z) is a member of the class Sb∗ . Therefore, noting that the distance d(1, ν) between the point 1 and the point Re(ν)−1 ν is cos(arg(b)) arg(b) = π2 , 3π or |Im(ν)| arg(b) = π2 or 3π , we can derive 2 2 the following two theorems. Theorem 2.3 If a function f (z) ∈ A satisfies the following inequality ∞ Re(ν) − 1 Re(ν) − 1 |an | |b| + |n − ν| + |b| + cos(arg(b)) − |1 − ν| cos(arg(b)) n=2 for some b ∈ C \ {0} (arg(b) = f (z) ∈ Sb∗ .

π 3π , ) 2 2

and a point ν given by (2.2), then

The proof is almost the same as that of Silverman’s result for Theorem 1.4. Proof. (2.3)

It follows that if f (z) satisfies Re(ν) − 1 zf (z) f (z) − ν < |b| + cos(arg(b)) ,

zf (z) maps U onto the interior region of the circle f (z) Cν . Therefore, it is sufficient that we prove if f (z) satisfies the inequality of the theorem, then the relation (2.3) holds true.

then f (z) ∈ Sb∗ , because

We know that

∞ n−1 (1 − ν) + (n − ν)a nz zf (z) n=2 = − ν ∞ f (z) n−1 1+ an z n=2

|1 − ν| +

n=2 ∞

1− |1 − ν| +

0, c = 1, 2, 3, · · · ),

∞ 1 {n − (1 − d)}|an | 1 d n=2

Corollary 2.6 :

f (z) ∈

=⇒

=⇒

f (z) ∈

"

∗ ∗ ∗ Sde iϕ ⊂ S−d ∩ Sd .

ϕ

In particular, if 0 < d 1, then ∞

Corollary 2.6 :

{n − (1 − d)}|an | 1 − (1 − d)

=⇒

f (z) ∈

[1+2d]

n=2

|an | +

1 d

∞

{n − (1 + d)} |an | 1

n=[1+2d]+1

where the symbol [ ] is defined by the following. [x] is the integer and satisfies x − 1 < [x] x for all x ∈ R.

∗ ∗ ∗ Sde iϕ ⊂ S−d ∩ S (1 − d).

ϕ

n=2

Corollary 5.3 :

"

=⇒

∗ f (z) ∈ S−d

60


When some concrete values are substituted for b, the following corollary is obtained. Corollary 5.8 For the case b = −

Corollary 2.6 :

2

1 2

(c = 1),

∞ n=2

Corollary 5.3 :

2

∞ n=2

For the case b = −1 Corollary 2.6 :

∞

1 n− |an | 1 2

=⇒

3 n− |an | 1 2

=⇒

f (z) ∈

ϕ

2

2

2

(c = 2),

n|an | 1

|a2 | +

S ∗1 eiϕ ⊂ S−∗ 1 ∩ S ∗

f (z) ∈ S−∗ 1 .

=⇒

f (z) ∈

"

n=2

Corollary 5.3 :

"

∗ Se∗iϕ ⊂ S−1 ∩ S∗.

ϕ ∞

(n − 2)|an | 1

=⇒

∗ f (z) ∈ S−1 .

n=3

For the case b = −

Corollary 2.6 :

Corollary 5.3 :

3 2

(c = 3),

∞ 2 1 |an | 1 n+ 3 n=2 2

=⇒

f (z) ∈

∞ 2 5 |a2 | + |a3 | + n− |an | 1 3 n=4 2

" ϕ

=⇒

S ∗3 eiϕ ⊂ S−∗ 3 . 2

2

f (z) ∈ S−∗ 3 . 2

1 2

.

61


For the case b = −2

(c = 4),

1 (n + 1)|an | 1 2 n=2 ∞

Corollary 2.6 :

=⇒

f (z) ∈

"

∗ ∗ S2e iϕ ⊂ S−2 .

ϕ

1 (n − 3)|an | 1 |a2 | + |a3 | + |a4 | + 2 n=5 ∞

Corollary 5.3 :

=⇒

∗ f (z) ∈ S−2 .

Example 5.9 For |ε| 12 , a function f (z) = z + εz 2 satisfies the inequality of Corollary 2.6 with b = −1, that is, f (z) ∈ S ∗ and for |ε| 1, satisfies that ∗ . of Corollary 5.3 with b = −1, that is, f (z) ∈ S−1 Actually, it follows that zf (z) 1 − 2|ε| 1 1 1 = 0 for all |ε| Re = Re 2 − > 2− f (z) 1 + εz 1 − |ε| 1 − |ε| 2 and 1 zf (z) 1 1 Re 1 + −1 = Re > > 0 for all |ε| 1. −1 f (z) 1 + εz 1 + |ε| ∗ (|ε| 1). Thus, f (z) ∈ S ∗ (|ε| 12 ) and f (z) ∈ S−1

Finally, the observation of the proof of Theorem 5.2, we can derive the similar results for the class Kb .

References [1] P. L. Duren, Univalent Functions, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1983. [2] T. Hayami, S. Owa and H. M. Srivastava, Coefficient inequalities for certain classes of analytic and univalent functions, J. Ineq. Pure Appl. Math. 8(4) Article 95 (2007), 1-10. [3] M. A. Nasr and M. K. Aouf, Starlike functions of complex order, J. Natural Sci. Math. 25 (1985), 1-12.

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[4] I. R. Nezhmetdinov and S. Ponnusamy, New coefficient conditions for the starlikeness of analytic functions and their applications, Houston J. Math. 31 (2005), 587-604. [5] M. S. Robertson, On the theory of univalent functions, Ann. Math. 37 (1936), 374-408. [6] H. Silverman, Univalent functions with negative coefficients, Proc. Amer. Math. Soc. 51 (1975), 109-116. [7] H. Silverman, E. M. Silvia, and D. Telage, Convolution conditions for convexity, starlikeness and spiral-likeness, Math. Z. 162 (1978), 125-130. Received: April, 2009