TAMKANG JOURNAL OF MATHEMATICS Volume 35, Number 3, Autumn 2004
NEW CLASSES OF k-UNIFORMLY CONVEX AND STARLIKE FUNCTIONS ESSAM AQLAN, JAY M. JAHANGIRI AND S. R. KULKARNI
Abstract. Certain classes of analytic functions are defined which will generalize new, as well as well-known, classes of k-uniformly convex and starlike functions. We provide necessary and sufficent coefficient conditions, distortion bounds, extreme points and radius of starlikeness for these classes.
1. Introduction Let A denote the family of functions f that are analytic in the open unit disc ∆ = {z : |z| < 1} and consider the subclassPT consisting of functions f in A which are univalent in ∞ ∆ and are of the form f (z) = z − m=2 am z m where am ≥ 0. The class T was introduced and studied by Silverman [9]. For 0 ≤ λ ≤ 1, 0 ≤ β < 1 and k ≥ 0 we let U(k, β, λ) consist of functions f in T satisfying the condition zf ′ (z) + λz 2 f ′′ (z) zf ′ (z) + λz 2 f ′′ (z) Re ≥ k − 1 + β. (1.1) ′ ′ (1 − λ)f (z) + λzf (z) (1 − λ)f (z) + λzf (z)
The family U(k, β, λ) is of special interest for it contains many well-known, as well as new, classes of analytic univalent functions. In particular, U(0, β, 0) is the family of functions starlike of order β and U(0, β, 1) is the family of functions convex of order β. For U(k, 0, 0) and U(k, 0, 1) we, respectively, obtain the classes of k- uniformly starlike and k-uniformly convex functions. The case for β to be other than zero, i.e. β ∈ (0, 1), is of special interest. For instance, if β ∈ (0, 1) then these classes are generalized to U(k, β, 0) and U(k, β, 1) of k-uniformly starlike functions of order β and k-uniformly convex functions of order β. More generally speaking, as β and λ vary, the family U(k, β, λ) provides a transition from the class k-uniformly starlike functions of order β and type λ to the class of k-uniformly convex functions of order β and type λ in ∆. The main feature of the elements of these classes is the fact that they map circular arcs with center at any point ζ in the open unit disk ∆ onto convex arcs or arcs starlike with respect to f (ζ), respectively. We remark that the classes of uniformly convex and uniformly starlike functions were introduced by Goodman [3,4] and later generalized by Kanas et Received April 11, 2001; revised September 10, 2001. 2000 Mathematics Subject Classification. Primary 30C45, secondary 30C50. Key words and phrases. Analytic, uniformly convex, starlike. 261
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al. in [5, 6, 7, 8]. In this paper we provide necessary and sufficent coefficient conditions, distortion bounds, extreme points, radius of starlikeness and convexity, closure theorem for functions in U(k, β, λ). 2. Main Results Our first theorem is on the necessary and sufficient coefficient requirements for functions to be in the class U(k, β, λ). Theorem 1. f ∈ U(k, β, λ) if and only if ∞ X
(1 + mλ − λ)(m(1 + k) − (k + β))am ≤ 1 − β
(2.1)
m=2
where 0 ≤ β < 1, k ≥ 2, 0 ≤ λ ≤ 1, and −π < θ ≤ π. Proof. We have f ∈ U(k, β, λ) if and only if the condition (1.1) is satisfied. Upon using the fact that Re w > k|w − 1| + β ⇔ Re{w(1 + keiθ ) − keiθ } > β the condition (1.1) may be written as zf ′ (z) + λz 2 f ′′ (z) iθ iθ ≥β Re (1 + ke ) − ke (1 − λ)f (z) + λzf ′ (z) or equivalenlty (zf ′ (z) + λz 2 f ′′ (z))(1 + keiθ ) − keiθ ((1 − λ)f (z) + λzf ′ (z)) Re ≥ β. (1 − λ)f (z) + λzf ′ (z)
(2.2)
Now we let A(z) = [zf ′ (z) + λz 2 f ′′ (z)](1 + keiθ ) − keiθ [(1 − λ)f (z) + λzf ′ (z)] and let B(z) = (1 − λ)f (z) + λzf ′ (z). Then (2.2) is equivalent to |A(z) + (1 − β)B(z)| ≥ |A(z) − (1 + β)B(z)| for 0 ≤ β < 1. For A(z) and B(z) as above, we have |A(z) + (1 − β)B(z)| = |(2 − β)z −
∞ X
(m + mλ(m − 1) + (1 − β)(1 − λ + mλ))am z m
m=2
−keiθ (
∞ X
[(m + mλ(m − 1) − (1 − λ + mλ)] am z m |
m=2
≥ (2−β)|z|−
∞ X
[m + mλ(m − 1) + (1 − β)(1 − λ + mλ)] am |z|m
m=2
−k
∞ X
m=2
[m + mλ(m − 2) − 1 + λ] am |z|m
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263
and, similarly, |A(z) − (1 + β)B(z)| ≤ β|z| +
∞ X
[m + mλ(m − 1) − (1 + β)(1 − λ + mλ)]am |z|m
m=2
+k
∞ X
[m + mλ(m − 1) − (1 − λ + mλ)]am |z|m .
m=2
Therefore, |A(z) + (1 − β)B(z)| − |A(z) − (1 + β)B(z)| ∞ X ≥ 2(1 − β)|z| − [2m + 2mλ(m − 1) − 2β(1 − λ + mλ)] m=2
−k[2m + 2mλ(m − 1) − 2(1 − λ + mλ)]am |z|m ≥ 0
or (1 − β) ≥
∞ X
[m(1 + k) + mλ(m − 1)(1 + k) − (1 − λ + mλ)(β + k)]am
m=2
which yields (1 − β) ≥
∞ X
(1 − λ + mλ)(m(1 + k) − (k + β)]am .
m=2
On the other hand, we must have [zf ′ (z) + λz 2 f ′′ (z)](1 + keiθ ) − keiθ [(1 − λ)f (z) + λzf ′ (z)] ≥ β. Re (1 − λ)f (z) + λzf ′ (z) Upon choosing the values of z on the positive real axis where 0 ≤ z = r < 1, the above inequality reduces to Re
(1 − β) − −keiθ
∞ X
[m − m2 λ − λ − β(1 − λ + mλ)]am rm−1
m=2 ∞ X
[m + m2 λ − λ − (1 − λ + mλ)]am rm−1
m=2
.
1−
∞ X
(1 − λ + mλ)am r
m=2
m−1
!
Since Re(−eiθ ) ≥ −|eiθ | = −1, the above inequality reduces to Re
(1 − β) −
∞ X
[m + m2 λ − λ − β(1 − λ + mλ)]am rm−1
m=2
≥ 0.
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ESSAM AQLAN, JAY M. JAHANGIRI AND S. R. KULKARNI
−k
∞ X
[m + m2 λ − λ − (1 − λ + mλ)]am rm−1
m=2
.
1−
∞ X
(1 − λ + mλ)am r
m−1
m=2
!
≥ 0.
Letting r → 1− we get desired conclusion. Remark. As special cases of Theorem 1, for λ = 0, see [2] and for k = 0, see [1]. The distortion theorem for the class U(k, β, λ) is given next. Theorem 2. If f ∈ U(k, β, λ) and |z| ≤ r < 1, then we have the sharp bounds r−
1−β 1−β r2 ≤ |f (z)| ≤ r + r2 (1 + λ)(2 + k − β) (1 + λ)(2 + k − β)
1−
2(1 − β) 2(1 − β) r ≤ |f ′ (z)| ≤ 1 + r. (1 + λ)(2 + k − β) (1 + λ)(2 + k − β)
and
(2.3)
Proof. We only prove the right hand side inequality in (2.3) since the other inequalities can be justified using similar arguments. On account of (2.1), we may write ∞ X
(1 + λ)(2 + k − β)am ≤
m=2
∞ X
(1 + mλ − λ)(m(1 + k) − (k + β))am ≤ 1 − β.
m=2
Hence |f (z)|≤|z| + |z|2
∞ X
am ≤r + r2
m=2
∞ X
am ≤r +
m=2
1−β r2 . (1 + λ)(2 + k − β)
The distortion bounds in Theorem 2 are sharp for f (z) = z −
(1 − β) z 2 , z = ±r. (1 + λ)(2 + k − β)
In the following theorem, we study the properties of extreme points of functions in the family U(k, β, λ). 1−β z m where λ ≥ 0, Theorem 3. Let f1 (z) = z and fm (z) = z − (1+mλ−λ)(m(1+k)−(k+β)) 0 ≤ β < 1, k ≥ 0, and m ≥ 2. Then f (z) is in U(k, β, λ) if and only if it can be expressed P∞ P∞ in the form f (z) = m=1 γm fm (z) where γm ≥ 0 and m=1 γm = 1. P∞ P∞ Proof. Let f (z) = m=1 γm fm (z) where γm ≥ 0 and m=1 γm = 1. Letting
f (z) = z −
∞ X
1−β γm z m . (1 + mλ − λ)(m(1 + k) − (k + β)) m=2
UNIFORMLY CONVEX AND STARLIKE FUNCTIONS
265
we get ∞ X (1 + mλ − λ)(m(1 + k) − (k + β)) 1−β γm 1−β (1 + mλ − λ)(m(1 + k) − (k + β)) m=2 X = γm = 1 − γ1 ≤ 1 (by Theorem 1). m=2
Therefore f ∈ U(k, β, λ). Conversely, suppose that f ∈ U(k, β, λ). Then am ≤
1−β , (1 + mλ − λ)(m(1 + k) − (k + β))
(m ≥ 2).
P Now, by letting γm = (1+mλ−λ)(m(1+k)−(k+β)) am and γ1 = 1 − ∞ m=2 γm we conclude 1−β P∞ P∞ the theorem, since f (z) = m=1 γm fm = γ1 f1 (z) + m=2 γm fm (z). Remark. For λ = 0, we obtain the extreme points given earlier in [2].
Finally, we discuss the radius of starlikeness of the functions in U(k, β, λ). Theorem 5. Let the f be in the class U(k, β, λ). Then f is starlike of order δ(0 ≤ δ < 1) in |z| < r2 (β, λ, k, δ), where r2 (β, λ, k, δ) = inf m
(1 − δ)(1 + mλ − λ)(m(1 + k) − (k + β)) (m − δ)(1 − β)
1 m−1
, m ≥ 2.
(2.4)
′ (z) Proof. It suffices to show that zff (z) − 1 ≤ 1 − δ for |z| < r2 (β, λ, k, δ). Note that P∞ ′ zf (z) − 1)am |z|m−1 ≤ m (m P − 1 . ∞ f (z) 1 − m am |z|m−1
′ (z) − 1 ≤ 1 − δ if we have the condition Now zff (z)
∞ X (m − δ)am |z|m−1 ≤ 1. (1 − δ) m=2
(2.5)
Considering the coefficient conditions required by Theorem 1, the above inequality (2.5) is true if m − δ m−1 (1 + mλ − λ)(m(1 + k) − (k + β)) |z| ≤ 1−δ (1 − β) or if |z| ≤
(1 − δ)(1 + mλ − λ)(m(1 + k) − (k + β)) (m − δ)(1 − β)
1 m−1
, m ≥ 2.
This last expression yields the bound required by the above theorem.
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ESSAM AQLAN, JAY M. JAHANGIRI AND S. R. KULKARNI
References [1] O. Altintas, On a subclass of certain starlike functions with negative coefficient, Math. Japon. 36 (1991), 489-495. [2] R. Bharati, R. Parvatham and A. Swaminathan, On subclasses of uniformaly convex functions and correspondding class of starlike functions, Tamkang J. Math. 28 (1997), 17-32. [3] A. W. Goodman, On uniformly convex functions, Ann. Polon. Math. 56 (1991), 87-92. [4] A. W. Goodman, On uniformly starlike functions, J. Math. Anal. Appl. 155 (1991), 364-370. [5] S. Kanas and A. Wisniownska, Conic regions and k-uniform convexity, J. Comput. Appl. Math. 105 (1999), 327-336. [6] S. Kanas and A. Wisniownska, Conic region and k-uniform convexity, II, Zeszyty. Nauk. Politech. Rzes. Mat. 22 (1998), 65-78. [7] S. Kanas and A. Wisniownska, Conic domains and starlike functions, Rev. Roumaine Math. Pures Appl. 45 (2000), 647-657. [8] S. Kanas and H. M. Srivastava, Linear operators associated with k-uniformly convex functions, Integral Transform Spec. Funct. 9 (2000), 121-132. [9] H. Silverman, Univalent functions with negative coefficients, Proc. Amer. Math. Soc. 51 (1975), 109-116.
Department of Mathematics, Fergusson College, Pune - 411 004, India. Mathematical Sciences, Kent State University, 14111 Claridon Troy Road, Burton, Ohio 440219500, U.S.A. E-mail:
[email protected] Department of Mathematics, Fergusson College, Pune - 411 004, India. E-mail: Kulkarni
[email protected]
