Integral properties of certain classes of analytic functions with negative coefficients

FUNCTIONS WITH NEGATIVE COEFFICIENTS

Received 20 February 2004 and in revised form 11 November 2004

We study integral properties of two classes of functions with negative coefficients defined using differential operators. The obtained results are sharp and they improve known re-sults.

1. Introduction

LetNdenote the set of nonnegative integers{0, 1,. . .,n,. . .},N∗_{=N\ {0}, and letᏺj,}

j∈N∗_{, be the class of functions of the form}

f(z)=z−

∞

k=j+1

akzk, ak≥0,k∈N,k≥j+ 1, (1.1)

that are analytic in the open unit discU= {z:|z|<1}. Definition 1.1[11]. The operatorDn_:ᏺ

j→ᏺj,n∈N, is defined by (a)D0f(z)= f(z);

(b)D1_{f(z)=D f(z)=z f(z); (c)D}n_f(z)=D(Dn−1_f(z)),z∈U.

Definition 1.2[4]. Letα,λ∈[0, 1),n∈N,j,m∈N∗_{; a function f belonging toᏺjis said} to be in the classTj(n,m,λ,α) if and only if

Re D

n+m_f(z)/Dn_f(z)

λDn+m_f(z)/Dn_{f(z)+ 1−λ}> α, z∈U. (1.2)

Remark 1.3. The classesTj(n,m,λ,α) are generalizations of the classes

(i)T1(0, 1, 0,α) andT1(1, 1, 0,α) defined and studied by Silverman [12] (these classes are the class of starlike functions with negative coefficients and the class of convex functions with negative coefficients, resp.),

(ii)Tj(0, 1, 0,α) andTj(1, 1, 0,α) studied by Chatterjea [7] and Srivastava et al. [13],

(iii)T1(n, 1, 0,α) studied by Hur and Oh [10],

(iv)T1(0, 1,λ,α) andT1(1, 1,λ,α) studied by Altintas and Owa [2], (v)T1(n, 1,λ,α) studied by Aouf and Cho [3,8],

(vi)T1(n,m, 0,α) studied by Hossen et al. [9].

Copyright©2005 Hindawi Publishing Corporation

In [4], the next characterization theorem of the classTj(n,m,λ,α) is given.

Theorem1.4. Letn∈N,j,m∈N∗_{,α,λ∈[0, 1), and let f ∈ᏺ}

j; then f ∈Tj(n,m,λ,α)

if and only if

∞

k=j+1

kn_km_{(1−αλ)−α(1−λ)a}

k≤1−α. (1.3)

The result is sharp and the extremal functions are

f(z)=z− 1−α

kn_km_{(1−αλ)−α(1−λ)}z

k_{, k∈N,k≥j+ 1. (1.4)}

Definition 1.5[5]. Letm,n∈N, j∈N∗_{,α∈[0, 1),λ∈[0, 1]; a function f belonging to} ᏺjis said to be in the classLj(n,m,λ,α) if and only if

Re(1−λ)D

n+1_{f(z) +λD}n+m+1_f(z)

(1−λ)Dn_{f(z) +λD}n+m_f(z) > α, z∈U. (1.5)

Remark 1.6. The classesLj(n,m,λ,α) are generalizations of the classes

(1)L1(0, 0, 0,α)=T1(0, 1, 0,α) andL1(1, 0, 1,α)=T1(1, 1, 0,α) (the classes defined and studied by Silverman [12]),

(2)Lj(0, 0, 0,α)=Tj(0, 1, 0,α) and Lj(0, 1, 1,α)= Lj(1, 0, 1,α)=Tj(1, 1, 0,α) (the

classes studied by Chatterjea [7] and Srivastava et al. [13]), (3)Lj(0, 1,λ,α) studied by Altintas [1],

(4)Lj(n, 1,λ,α),Lj(n,m, 0,α), andLj(n, 1, 1,α) studied by Aouf and Srivastava [6].

In [5], the next characterization theorem of the classLj(n,m,λ,α) is given.

Theorem 1.7. Let n,m∈N, j∈N∗_{, α∈[0, 1), λ∈[0, 1], and let f ∈ᏺj; then f ∈}

Lj(n,m,λ,α)if and only if

∞

k=j+1

kn(k−α)1 +km−1λak≤1−α. (1.6)

The result is sharp and the extremal functions are

f(z)=z− 1−α

kn_{(k−α)1 +k}m_−1λz

k_{, k∈N,k≥j+ 1. (1.7)}

LetIc:ᏺj→ᏺjbe the integral operator defined byg=Ic(f), wherec∈(−1,∞), f ∈

ᏺj, and

g(z)=c+ 1 zc

_z

0t

c−1_{f(t)dt. (1.8)}

We note that if f ∈ᏺjis a function of the form (1.1), then

g(z)=Ic(f)(z)=z−

∞

k=j+1

c+ 1 c+kakz

By usingTheorem 1.4, in [4] it is proved thatIc(Tj(n,m,λ,α))⊂Tj(n,m,λ,α) and by

usingTheorem 1.7, in [5] it is proved thatIc(Lj(n,m,λ,α))⊂Lj(n,m,λ,α). In this note,

these results are improved.

2. Integral properties of the classTj(n,m,λ,α)

Theorem 2.1. Let n∈N, j,m∈N∗_{,α,λ∈[0, 1), and letc∈(−1,∞); if f ∈T}

j(n,m,

λ,α)andg=Ic(f), theng∈Tj(n,m,λ,β), where

β=β(m,λ,α,c;j+ 1)

=1−

(j+ 1)m_{−1(1−α)(1−λ)(c+ 1)}

(j+ 1)m_{−1(1−αλ)(c+j+ 1)−λ(c+ 1)(1−α)+ (1−α)j}

(2.1)

andα < β(m,λ,α,c;j+ 1)<1. The result is sharp.

Proof. FromTheorem 1.4and from (1.9) we haveg∈Tj(n,m,λ,β) if and only if

∞

k=j+1

kn_km_{(1−βλ)−β(1−λ)(c+ 1)}

(1−β)(c+k) ak≤1. (2.2)

We find the largestβsuch that (2.2) holds. We note that the inequalities

km_{(1−βλ)−β(1−λ)}

1−β

c+ 1 c+k ≤

km_{(1−αλ)−α(1−λ)}

1−α , k≥j+ 1, (2.3)

imply (2.2), because f ∈Tj(n,m,λ,α) and it satisfies (1.3). But the inequalities (2.3) are

equivalent to

A(m,λ,α,c;k)β≤B(m,λ,α,c;k), (2.4)

where

A(m,λ,α,c;k)=km−1(1−αλ)(c+k)−λ(c+ 1)(1−α)+ (1−α)(k−1),

B(m,λ,α,c;k)=A(m,λ,α,c;k)−km−1(c+ 1)(1−α)(1−λ). (2.5)

Since 1−αλ >1−αandc+k > c+ 1, we haveA(m,λ,α,c;k)>0 and from (2.4) we obtain

β≤B(m,λ,α,c;k)

We define β(m,λ,α,c;k) :=B(m,λ,α,c;k)/A(m,λ,α,c;k). We show now that β(m,λ,α, c;k) is an increasing function ofk,k≥j+ 1. Indeed

β(m,λ,α,c;k)=1−(1−α)(1−λ)(c+ 1) km−1 A(m,λ,α,c;k)

=1−(1−α)(1−λ)(c+ 1) 1 E(m,λ,α,c;k),

(2.7)

where E(m,λ,α,c;k)=A(m,λ,α,c;k)/(km_{−1) and β(m,λ,α,c;k) increases whenk}

in-creases if and only ifE(m,λ,α,c;k) is also an increasing function ofk. Leth(x)=E(m,λ,α,c;x),x∈[j+ 1,∞)⊂[2,∞); we have

h(x)=1−αλ+ (1−α)x

m_−1−mxm_+xm−1

xm₋₁2

=1−αλ+ (1−α)

_1−m xm₋₁+

mxm−1₋₁

xm₋₁2

>1−αλ−(1−α)=α(1−λ)≥0, x∈[j+ 1,∞),

(2.8)

where we used the fact that

1−m xm₋₁+

mxm−1₋₁

xm₋₁2 ≥

1−m

xm₋₁>−1. (2.9)

We obtainedh(j+ 1)≤h(k),k≥j+ 1, and this implies

β=β(m,λ,α,c;j+ 1)≤β(m,λ,α,c;k), k≥j+ 1. (2.10)

The result is sharp because

Ic

fα

= fβ, (2.11)

where

fα(z)=z− 1−α

(j+ 1)n_{(j+ 1)}m_{(1−αλ)−α(1−λ)}zj

+1_,

fβ(z)=z−

1−β

(j+ 1)n_{(j+ 1)}m_{(1−βλ)−β(1−λ)}z j+1

(2.12)

are the extremal functions ofTj(n,m,λ,α) andTj(n,m,λ,β), respectively, andβ=β(m,λ,

α,c;j+ 1).

Indeed, we have

Ic

fα

(z)=z− (1−α)(c+ 1)

(j+ 1)n_{(c+j+ 1)(j+ 1)}m_{(1−αλ)−α(1−λ)}zj

But if we use the notationsA=A(m,λ,α,c;j+ 1) andB=B(m,λ,α,c;j+ 1), we deduce 1−β

(j+ 1)m_{(1−βλ)−β(1−λ)}

= A−B

(j+ 1)m_{(A−Bλ)−B(1−λ)}

=

(j+ 1)m_{−1(1−α)(1−λ)(c+ 1)}

(1−λ) A(j+ 1)m_{+(j+ 1)}m_{−1λ(j+ 1)}m_{(1−α)(c+ 1)−B}

=

(j+ 1)m_{−1(1−α)(c+ 1)}

(j+ 1)m_{−1(j+ 1)}m_{λ(1−α)(1 +c) +A+ (1−α)(c+ 1)(1−λ)}

= (1−α)(c+ 1)

(c+j+ 1)(j+ 1)m_{(1−αλ)−α(1−λ)}

(2.14)

and this implies (2.11).

Fromβ=1−[(j+ 1)m_{−1](1−α)(1−λ)(c+ 1)/Aand becauseA >0, we obtainβ <1.}

We also haveβ > α; indeed

β−α=(1−α)

1−

(j+ 1)m_{−1(c+ 1)(1−λ)}

(j+ 1)m_{−1(1−αλ)(c+j+ 1)−λ(c+ 1)(1−α)+ (1−α)j}

>(1−α)

1− (c+ 1)(1−λ)

(1−αλ)(c+j+ 1)−λ(c+ 1)(1−α)

= (1−α)(1−αλ)j

j(1−αλ) + (c+ 1)(1−λ)>0.

(2.15)

3. Integral properties of the classLj(n,m,λ,α)

Theorem 3.1. Let n,m∈N, j∈N∗_{,α∈[0, 1),λ∈[0, 1], and letc∈(−1,∞); if f ∈}

Lj(n,m,λ,α)andg=Ic(f), theng∈Lj(n,m,λ,γ), where

γ=γ(α,c;j+ 1)=1−(1−α)(c+ 1)

2−α+c+j (3.1)

andα < γ(α,c;j+ 1)<1. The result is sharp.

Proof. FromTheorem 1.7and from (1.9) we haveg∈Lj(n,m,λ,β) if and only if

∞

k=j+1

kn_{(k−γ)1 +k}m_{−1λ(c+ 1)}

(1−γ)(c+k) ak≤1. (3.2)

We find the largestγsuch that (3.2) holds. We note that the inequalities (k−γ)(c+ 1)

(1−γ)(c+k)≤ k−α

imply (3.2), because f ∈Lj(n,m,λ,α). But the inequalities (3.3) are equivalent to

(k−1)(k+c+ 1−α)γ≤(k−1)(k+αc), k≥j+ 1. (3.4)

Since (k+c+ 1−α)>0 andk−1≥j≥1, we deduce

γ≤ k+αc

k+c+ 1−α ∀k≥j+ 1. (3.5)

We define γ(α,c;k) :=1−(1−α)(c+ 1)/(k+c+ 1−α). Obviously, γ(α,c;j+ 1)≤γ(α, c;k) fork≥j+ 1, hence we obtain thatγ=γ(α,c;j+ 1).

We haveγ <1 because (1−α)(c+ 1)/(k+c+ 1−α)>0 andγ > αbecause

γ−α=(1−α) 1−α+j

2−α+c+j >0. (3.6)

The result is sharp. Indeed, we consider the function

ϕα(z)=z− 1−α

(j+ 1)n_{(j+ 1−α)1−λ+λ(j+ 1)}mz

j+1 _(3.7)

that belongs toLj(n,m,λ,α). Then

Ic(ϕα)(z)=z− (1−α)(c+ 1)

(j+ 1)n_{(j+ 1−α)1−λ+λ(j+ 1)}m_{(c+j+ 1)}z

j+1_{, (3.8)}

and because

(1−α)(c+ 1) (j+ 1−α)(c+j+ 1)=

1−γ

j+ 1−γ, (3.9)

we deduce thatIc(ϕα)=ϕγbelongs toLj(n,m,λ,γ).

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G. S. Sˇalˇagean: Faculty of Mathematics and Computer Science, Babes-Bolyai University, 1 M. Ko-galniceanu Street, 3400 Cluj-Napoca, Romania