# Pascu-Type Harmonic Functions with Positive Coefficients Involving Salagean Operator

Authors: K. Vijaya; G. Murugusundaramoorthy; M. Kasthuri
Journal: International Journal of Analysis
 (2014)

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## Abstract

Making use of a Salagean operator, we introduce a new class of complex valued
harmonic functions which are orientation preserving and univalent in the open unit disc. Among the results presented in this paper including the coeffcient bounds, distortion inequality, and covering property, extreme points, certain inclusion results, convolution properties, and partial sums for this generalized class of functions are discussed.


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## Body

## 1. Introduction and Preliminaries

A continuous functionf=u+iv is a complex-valued harmonic function in a complex domain 𝒢 if both u and v are real and harmonic in 𝒢. In any simply connected domain 𝒟⊂𝒢, we can write f=h+g¯, where h and g are analytic in 𝒟. We call h the analytic part and g the coanalytic part of f. A necessary and sufficient condition for f to be locally univalent and orientation preserving in 𝒟 is that |h′(z)|>|g′(z)| in 𝒟 (see [1]).Denote byℋ the family of functions
(1)f=h+g¯
which are harmonic, univalent, and orientation preserving in the open unit disc 𝒰={z:|z|<1} so that f is normalized by f(0)=fz(0)-1=0. Thus, for f=h+g¯∈ℋ, the functions h and g are analytic in 𝒰 and can be expressed in the following forms:
(2)h(z)=z+∑n=2∞‍anzn,g(z)=∑n=1∞‍bnzn(0≤b1<1),
and f(z) is then given by
(3)f(z)=z+∑n=2∞‍anzn+∑n=1∞‍bnzn¯(0≤|b1|<1).
We note that the family ℋ of orientation preserving, normalized harmonic univalent functions reduces to the well-known class 𝒮 of normalized univalent functions if the coanalytic part of f is identically zero; that is, g≡0.For functionsf∈ℋ, Jahangiri et al. [2] defined Salagean operator on harmonic functions given by
(4)Dℓf(z)=Dℓh(z)+(-1)ℓDℓg(z)¯,
where
(5)Dℓh(z)=z+∑n=2∞‍nℓanzn,Dℓg(z)=∑n=1∞‍nℓbnzn.In 1975, Silverman [3] introduced a new class 𝒯 of analytic functions of the form f(z)=z-∑n=2∞|an|zn and opened up a new direction of studies in the theory of univalent functions as well as in harmonic functions with negative coefficients [4]. Uralegaddi et al. [5] introduced analogous subclasses of star-like, convex functions with positive coefficients and opened up a new and interesting direction of research. In fact, they considered the functions where the coefficients are positive rather than negative real numbers. Motivated by the initial work of Uralegaddi et al. [5], many researchers (see [6–9]) introduced and studied various new subclasses of analytic functions with positive coefficients but analogues results on harmonic univalent functions have not been explored in the literature. Very recently, Dixit and Porwal [10] attempted to fill this gap by introducing a new subclass of harmonic univalent functions with positive coefficients.Denote by𝒱ℋ the subfamily of ℋ consisting of harmonic functions f=h+g¯ of the form
(6)f(z)=z+∑n=2∞‍anzn-∑n=1∞‍bnzn¯(an≥0;bn≥0,0≤|b1|<1).Motivated by the earlier works of [11–14] on the subject of harmonic functions, in this paper an attempt has been made to study the class of functions f∈𝒱ℋ associated with Salagean operator on harmonic functions. Further, we obtain a sufficient coefficient condition for functions f∈ℋ given by (3) and also show that this coefficient condition is necessary for functions f∈𝒱ℋ, the class of harmonic functions with positive coefficients. Distortion results and extreme points, inclusion relations, and convolution properties and results on partial sums are discussed extensively.For0≤λ≤1, 1<γ≤4/3, we let 𝒫ℋℓ(λ,γ) be a new subclass of ℋ, consisting of all functions of the form (3) satisfying the condition
(7)ℜ((1-λ)Dℓ+1f(z)+λDℓ+2f(z)(1-λ)Dℓf(z)+λDℓ+1f(z))<γ,
where Dℓf(z) is given by (4) (see [2]). Also let 𝒱ℋℓ(λ,γ)=𝒫ℋℓ(λ,γ)⋂‍𝒱ℋ.


## 2. Coefficient Bounds

In our first theorem, we obtain a sufficient coefficient condition for harmonic functions in𝒫ℋℓ(λ,γ).Theorem 1.
Letf=h+g¯ be given by (3). If
(8)∑n=2∞‍nℓ|1-λ+nλ|(n-γ)γ-1|an|+∑n=1∞‍nℓ|1-λ-nλ|(n+γ)γ-1|bn|≤1,
where a1=1 and 1<γ≤4/3, then f∈𝒫ℋℓ(λ,γ).Proof.
We let (8) hold for the coefficients of f=h+g¯. It suffices to show that
(9)|A(z)/B(z)-1A(z)/B(z)-(2γ-1)|<1,z∈𝒰,
where
(10)A(z)=(1-λ)Dℓ+1f(z)+λDℓ+2f(z)=z+∑n=2∞‍nℓ+1(1-λ+nλ)anzn-(-1)ℓ∑n=1∞‍nℓ+1(1-λ-nλ)bn¯z¯n,(11)B(z)=(1-λ)Dℓf(z)+λDℓ+1f(z)=z+∑n=2∞‍nℓ(1-λ+nλ)anzn+(-1)ℓ∑n=1∞‍nℓ(1-λ-nλ)bn¯z¯n.
Substituting for A(z) and B(z) in (9), we get
(12)|A(z)/B(z)-1A(z)/B(z)-(2γ-1)|≤(∑n=2∞nℓ|1-λ+nλ|(n-1)|an||z|n-1+(-1)ℓ00000×∑n=1∞‍nℓ|1-λ-nλ|(n+1)|bn||z|n-1)×(2(γ-1)-∑n=2∞‍nℓ|1-λ+nλ|[n-(2γ-1)]00000000×|an||z|n-1-(-1)ℓ00000000×∑n=1∞‍nℓ|1-λ-nλ|[n+(2γ-1)]|bn||z|n-1)-1≤(∑n=2∞‍nℓ|1-λ+nλ|(n-1)|an|000000+(-1)ℓ∑n=1∞‍nℓ|1-λ-nλ|(n+1)|bn|)×(2(γ-1)-∑n=2∞‍nℓ|1-λ+nλ|[n-(2γ-1)]|an|00000000-(-1)ℓ∑n=1∞‍nℓ|1-λ-nλ|[n+(2γ-1)]|bn|)-1.
The above expression is bounded above by 1 if(13)2∑n=2∞‍nℓ|1-λ+nλ|(n-1)|an|+2∑n=1∞‍nℓ|1-λ-nλ|(n+1)|bn|≤2(γ-1)
which is equivalent to
(14)∑n=2∞‍nℓ|1-λ+nλ|(n-γ)γ-1|an|+∑n=1∞‍nℓ|1-λ-nλ|(n+γ)γ-1|bn|≤1.
But (8) is true by hypothesis. Hence, |(A(z)/B(z)-1)/(A(z)/B(z)-(2γ-1))|<1, z∈𝒰, and the theorem is proved for A(z) and B(z) is given by (10) and (11), respectively.Theorem 2.
Fora1=1 and 1<γ≤4/3, f=h+g¯∈𝒱ℋℓ(λ,γ) if and only if
(15)∑n=2∞‍nℓ|1-λ+nλ|(n-γ)γ-1|an|+∑n=1∞‍nℓ|1-λ-nλ|(n+γ)γ-1|bn|≤1.Proof.
Since𝒱ℋℓ(λ,γ)⊂𝒫ℋℓ(λ,γ), we only need to prove the “only if” part of the theorem. To this end, for functions f of the form (6), we notice that the condition
(16)ℜ((1-λ)Dℓ+1f(z)+λDℓ+2f(z)(1-λ)Dℓf(z)+λDℓ+1f(z))<γ.
Equivalently,
(17)ℜ((∑n=2∞‍-1((1-γ)z-∑n=2∞‍nℓ(1-λ+nλ)(n-γ)anzn000000-(-1)2ℓ∑n=1∞‍nℓ(1-λ-nλ)(n+γ)bnz¯n)000×(z-∑n=2∞‍nℓ(1-λ+nλ)anzn00000000+(-1)2ℓ∑n=1∞‍nℓ(1-λ-nλ)bnz¯n)-1)≥0.
The above required condition must hold for all values of z in 𝒰. Upon choosing the values of z on the positive real axis where 0≤z=r<1, we must have
(18)(1-γ-∑n=2∞‍nℓ|1-λ+nλ|(n-γ)anrn-10000-∑n=1∞‍nℓ|1-λ-nλ|(n+γ)bnrn-1)×(1-∑n=2∞‍nℓ|(1-λ+nλ)|anrn-1000000+∑n=1∞‍nℓ|1-λ-nλ|bnrn-1)-1≥0.
If condition (15) does not hold, then the numerator in (18) is negative for r sufficiently close to 1. Hence, there exists z0=r0 in (0,1) for which the quotient of (18) is negative. This contradicts the required condition for f∈𝒱ℋℓ(λ,γ). This completes the proof of the theorem.


## 3. Distortion Bounds and Extreme Points

By routine procedure (see [10–13]),  we can easily prove the following results; hence we state the following theorems without proof for functions in 𝒱ℋℓ(λ,γ).Theorem 3 (distortion bounds).
Letf∈𝒱ℋℓ(λ,γ). Then for |z|=r<1, we have
(19)(1-|b1|)r-12ℓ(1+λ)(γ-12-γ-1+γ2-γ|b1|)r2≤|f(z)|≤(1+|b1|)r+12ℓ(1+λ)(γ-12-γ-1+γ(2-γ)|b1|)r2.Corollary 4 (covering result).
Iff(z)∈𝒱ℋℓ(λ,γ), then
(20){w:|w|<2ℓ+1(1+γ)+1-[2ℓ(1+γ)+1]γ2ℓ(2-γ)(1+λ)000-2ℓ+1(1+γ)-1-[2ℓ(1+γ)+1]γ2ℓ(2-γ)(1+λ)}b1⊂f(U).Next we state the extreme points of closed convex hulls of𝒱ℋℓ(λ,γ) denoted by clco⁡𝒱ℋℓ(λ,γ).Theorem 5.
A functionf(z)∈𝒱ℋℓ(λ,γ) if and only if f(z)=∑n=1∞(Xnhn(z)+Yngn(z)) where h1(z)=z,  hn(z)=z+((γ-1)/(2ℓ|1-λ+nλ|(n-γ)))zn, (n≥2), and gn(z)=z+((γ-1)/(2ℓ|1-λ-nλ|(n+γ)))z¯n, (n≥1) also ∑n=1∞(Xn+Yn)=1, Xn≥0 and Yn≥0. In particular, the extreme points of 𝒱ℋℓ(λ,γ) are {hn} and {gn}.Theorem 6.
The family𝒱ℋℓ(λ,γ) is closed under convex combinations.


## 4. Inclusion Results

Now, we will examine the closure properties of the class𝒱ℋℓ(λ,γ) under the generalized Bernardi-Libera-Livingston integral operator ℒc(f) which is defined by ℒc(f)=((c+1)/zc)∫0ztc-1f(t)dt,  c>-1.Theorem 7.
Letf(z)∈𝒱ℋℓ(λ,γ). Then ℒc(f(z))∈𝒱ℋℓ(λ,γ).Lemma 8 (see [15]).
Letf=h+g¯ be given by (3). If
(21)∑n=2∞‍2ℓ|1-λ+nλ|(n-α)1-α|an|+∑n=1∞‍2ℓ|1-λ-nλ|(n+α)1-α|bn|≤1,
where a1=1 and 0≤α<1, then f∈ℳℋ*(λ,α).Theorem 9.
Letf=h+g¯∈𝒱ℋℓ(λ,α) be given by (6). Then f∈𝒱ℋℓ(λ,(4-3γ)/(3-2γ)).Proof.
Sincef∈𝒱ℋℓ(λ,γ), then by Theorem 1 we must have
(22)∑n=2∞‍2ℓ|1-λ+nλ|(n-γ)γ-1|an|+∑n=1∞‍2ℓ|1-λ-nλ|(n+γ)γ-1|bn|≤1.
To show that f∈𝒱ℋℓ(λ,(4-3γ)/(3-2γ)), by virtue of Lemma 8 we have to show that
(23)∑n=2∞‍2ℓ|1-λ+nλ|[n-((4-3γ)/(3-2γ))]1-((4-3γ)/(3-2γ))|an|+∑n=1∞‍2ℓ|1-λ-nλ|[n+((4-3γ)/(3-2γ))]1-((4-3γ)/(3-2γ))|bn|≤1,
where 0≤(4-3γ)/(3-2γ)<1. For this, it is sufficient to prove that
(24)|1-λ+nλ|(n-γ)γ-1≥|1-λ+nλ|[n-((4-3γ)/(3-2γ))]1-((4-3γ)/(3-2γ)),0000000000000000000000000(n=2,3,…),|1-λ-nλ|(n+γ)γ-1≥|1-λ-nλ|[n+((4-3γ)/(3-2γ))]1-((4-3γ)/(3-2γ)),000000000000000000000000(n=1,2,3,…)
or equivalently (γ-1)|1-λ+nλ|(n-γ)≥0(n=2,3,…), and (γ-1)|1-λ-nλ|(n+γ)≥0(n=1,2,3,…), which is true and the theorem is proved.Corollary 10.
𝒱
ℋ
ℓ
(
λ
,
γ
)
⊂
𝒱
ℋ
ℓ
(
λ
,
4
/
3
)
⊂
𝒱
ℋ
ℓ.


## 5. Convolution Properties

For functionsf∈ℋ given by (3) and F∈ℋ given by
(25)F(z)=H(z)+G(z)¯=z+∑n=2∞‍Anzn+∑n=1∞‍Bnzn¯,
we recall the Hadamard product (or convolution) of f and F by
(26)(f*F)(z)=z+∑n=2∞‍anAnzn+∑n=1∞‍bnBnzn¯(z∈𝒰).
Let Fj(z)∈𝒱ℋℓ(λ,γ)(j=1,2,3,…,p) be given by
(27)Fj(z)=z+∑n=2∞‍|an,j|zn+(-1)ℓ∑n=1∞‍|bn,j|zn¯,
then the convolution is defined by
(28)(F1*⋯*Fp)(z)=z-∑n=2∞‍∏j=1p‍|an,j|zn+(-1)ℓ∑n=1∞‍∏j=1p‍|bn,j|zn¯.Theorem 11.
LetFj(z)∈𝒱ℋℓ(λ,γj)(j=1,2,3,…,p), then (F1*⋯*Fp)(z)∈𝒱ℋℓ(λ,β), where
(29)β=1+∏j=1p(γj-1)2ℓ(1+λ)∏j=1p(2+γj)-∏j=1p(γj-1).Proof.
We use the principle of mathematical induction in our proof. LetF1∈𝒱ℋℓ(λ,γ1), and F2∈𝒱ℋℓ(λ,γ2). By using Theorem 2, we have
(30)∑n=2∞‍nℓ|1-λ+nλ|(n-γj)γj-1|an,j|+∑n=1∞‍nℓ|1-λ-nλ|(n+γj)γj-1|bn,j|≤1;
then
(31)[∑n=2∞‍(nℓ|1-λ+nλ|(n-γ1)γ1-1|an,1|)200×∑n=2∞‍(nℓ|1-λ+nλ|(n-γ2)γ2-1|an,2|)2]1/2+[∑n=1∞‍(nℓ|1-λ-nλ|(n+γ1)γ1-1|bn,1|)200000000×∑n=1∞‍(nℓ|1-λ-nλ|(n+γ2)γ2-1|bn,2|)2]1/2≤1.
Thus, by applying Cauchy-Schwarz inequality, we have
(32)∑n=2∞‍(n2ℓ|1-λ+nλ|(n-γ1)(n-γ2)(γ1-1)(γ2-1)|an,1an,2|)2≤[∑n=2∞‍(nℓ|1-λ+nλ|(n-γ1)γ1-1|an,1|)2]1/2×[∑n=2∞‍(nℓ|1-λ+nλ|(n-γ2)γ2-1|an,2|)2]1/2,∑n=1∞‍(n2ℓ|1-λ-nλ|(n+γ1)(n+γ2)(γ1-1)(γ2-1)|bn,1bn,2|)2≤[∑n=1∞‍(nℓ|1-λ+nλ|(n+γ1)γ1-1|bn,1|)2]1/2×[∑n=1∞‍(nℓ|1-λ+nλ|(n+γ2)γ2-1|bn,2|)2]1/2.
Then, we get
(33)∑n=2∞‍nℓ|1-λ+nλ|(n-γ1)(n-γ2)(γ1-1)(γ2-1)|an,1an,2|+∑n=1∞‍n2ℓ|1-λ-nλ|(n+γ1)γ1-1|bn,1bn,2|≤1.
Therefore, if
(34)∑n=2∞‍nℓ|1-λ+nλ|(n-α)α-1|an,1an,2|≤∑n=2∞‍n2ℓ|1-λ+nλ|(n-γ1)(n-γ2)(γ1-1)(γ2-1)|an,1an,2|,∑n=1∞‍nℓ|1-λ-nλ|(n+α)α-1|bn,1bn,2|≤∑n=1∞‍n2ℓ|1-λ-nλ|(n+γ1)(n+γ2)(γ1-1)(γ2-1)|bn,1bn,2|,
that is, if
(35)|an,1an,2|≤α-1n-α|1-λ+nλ|(n-γ1)(n-γ2)(γ1-1)(γ2-1),|bn,1bn,2|≤α-1n+α|1-λ-nλ|(n+γ1)(n+γ2)(γ1-1)(γ2-1),
then (F1*F2)(z)∈𝒱ℋℓ(λ,α). By (30) we have
(36)∑n=2∞‍nℓ|1-λ+nλ|(n-γj)γj-1|an,j|≤1,∑n=1∞‍nℓ|1-λ-nλ|(n+γj)γj-1|bn,j|≤1.
Hence we get
(37)|an,j|≤γj-1|1-λ+nλ|(n-γj)|bn,j|≤γj-1|1-λ-nλ|(n+γj).
Consequently, if
(38)(γ1-1)(γ2-1)n2ℓ|1-λ+nλ|2(n-γ1)(n-γ2)≤α-1n-α|1-λ+nλ|(n-γ1)(n-γ2)(γ1-1)(γ2-1),(γ1-1)(γ2-1)n2ℓ|1-λ-nλ|2(n+γ1)(n+γ2)≤α-1n+α|1-λ-nλ|(n+γ1)(n+γ2)(γ1-1)(γ2-1).
That is, if
(39)n-αα-1≤|1-λ+nλ|(n-γ1)(n-γ2)(γ1-1)(γ2-1),n+αα-1≤|1-λ-nλ|(n+γ1)(n+γ2)(γ1-1)(γ2-1),
then (F1*F2)(z)∈𝒱ℋℓ(λ,α). Then we see that
(40)α≥1+(n-1)(γ1-1)(γ2-1)nℓ|1-λ+nλ|(n+γ1)(n+γ2)+(γ1-1)(γ2-1)=ϕ(n),α≥1+(n+1)(γ1-1)(γ2-1)nℓ|1-λ-nλ|(n+γ1)(n+γ2)-(γ1-1)(γ2-1)=ψ(n).
Since ϕ(n) for n≥2 and ψ(n) for n≥1 are increasing,
(41)α≥1+(γ1-1)(γ2-1)2ℓ|1+λ|(2+γ1)(2+γ2)+(γ1-1)(γ2-1)(42)α≥1+2(γ1-1)(γ2-1)|1-2λ|(1+γ1)(1+γ2)-(γ1-1)(γ2-1),
and also
(43)(γ1-1)(γ2-1)2ℓ|1+λ|(2+γ1)(2+γ2)+(γ1-1)(γ2-1)≤2(γ1-1)(γ2-1)|1-2λ|(1+γ1)(1+γ2)-(γ1-1)(γ2-1),
then (F1*F2)(z)∈𝒱ℋℓ(λ,α) where
(44)α≥1+(γ1-1)(γ2-1)2ℓ(1+λ)(2+γ1)(2+γ2)+(γ1-1)(γ2-1).
Next, we suppose that (F1*F2*⋯*Fp)(z)∈𝒱ℋℓ(λ,β), where
(45)β=1+∏j=1p(γj-1)2ℓ(1+λ)∏j=1p(2+γj)-∏j=1p(γj-1).
We can show that(F1*F2*⋯*Fp+1)(z)∈𝒱ℋℓ(λ,δ), where
(46)δ≥1+(β-1)(γp+1-1)2ℓ(1+λ)(2+β)(2+γp+1)+(β-1)(γp+1-1).
Since
(47)(β-1)(αp+1-1)=∏j=1p+1(γj-1)2ℓ(1+λ)∏j=1p(2+γj)-∏j=1p(γj-1),(β-2)(αp+1-2)=∏j=1p+1(γj-1)2ℓ(1+λ)∏j=1p(2+γj)-∏j=1p(γj-1),
we have
(48)δ=1+∏j=1p+1(γj-1)2ℓ(1+λ)∏j=1p+1(2+γj)-∏j=1p+1(γj-1).Corollary 12.
LetFj(z)∈𝒱ℋℓ(λ,γ)(j=1,2,3,…,p), then (F1*⋯*Fp)(z)∈𝒱ℋℓ(λ,β), where
(49)β=1+(γ-1)p2ℓ(1+λ)(2+γ)p-(γ-1)p.


## 6. Partial Sums Results

In 1985, Silvia [16] studied the partial sums of convex functions of order α(0≤α<1). Later on, Silverman [17] and several researchers studied and generalized the results on partial sums for various classes of analytic functions only but analogues results on harmonic functions have not been explored in the literature. Very recently, Porwal [18] and Porwal and Dixit [19] filled this gap by investigating interesting results on the partial sums of star-like harmonic univalent functions. Now in this section we discussed the partial sums results for the class of harmonic functions with positive coefficients based on Salagean operator of order γ(1<γ≤4/3) on lines similar to Porwal [18].Let𝒫ℋℓ(An/δ,Bn/δ) denote the subclass of ℋ consisting of functions f=h+g¯ of the form (3) which satisfy the inequality
(50)∑n=2∞‍Anδ|an|+∑n=1∞‍Bnδ|bn|≤1,
where
(51)Anδ=nℓ|1-λ+nλ|(n-γ)γ-1,Bnδ=nℓ|1-λ-nλ|(n+γ)γ-1
and δ=γ-1, unless otherwise stated.Now, we discuss the ratio of a function of the form (6) with b1=0 being
(52)fm(z)=z+∑n=2m‍anzn+∑n=1∞‍bnzn¯,fk(z)=z+∑n=2∞‍anzn+∑n=2k‍bnzn¯,fm,k(z)=z+∑n=2m‍anzn+∑n=2k‍bnzn¯.We first obtain the sharp bounds forℜ{f(z)/fm(z)}.Theorem 13.
Iff of the form (6) with b1=0 satisfies the condition (50), then
(53)ℜ{f(z)fm(z)}≥Am+1-δAm+1,(z∈𝒰),
where
(54)An≥{δ,ifn=2,3,…,m,Am+1,ifn=m+1,m+2,…,Bn≥δ,ifn=2,3,….
The result (53) is sharp with the function given by
(55)f(z)=z+δAm+1zm+1.Proof.
Define the functionw(z) by
(56)1+w(z)1-w(z)=Am+1δ[f(reiθ)fm(reiθ)-Am+1-δAm+1]=((∑n=m+1∞‍anrn-1ei(n-1)θ)1+∑n=2m‍anrn-1ei(n-1)θ00000+∑n=2∞‍bn¯rn-1e-i(n+1)θ00000-Am+1δ(∑n=m+1∞‍anrn-1ei(n-1)θ))×(1+∑n=2m‍anrn-1ei(n-1)θ000000+∑n=2∞‍bn¯rn-1e-i(n+1)θ)-1.
It suffices to show that |w(z)|≤1. Now, from (56), we can write
(57)w(z)=(Am+1δ(∑n=m+1∞‍anrn-1ei(n-1)θ))×((∑n=m+1∞‍anrn-1ei(n-1)θ)2+2(∑n=2∞‍anrn-1ei(n-1)θ000000000000+∑n=2∞‍bn¯rn-1e-i(n+1)θ)000000+Am+1δ(∑n=m+1∞‍anrn-1ei(n-1)θ))-1.
Hence we obtain
(58)|w(z)|≤(Am+1/δ)(∑n=m+1∞|an|)2-2[∑n=2m|an|+∑n=2∞|bn|]-(Am+1/δ)∑n=m+1∞|an|.
Now |w(z)|≤1 if
(59)∑n=2∞‍|an|+∑n=2∞‍|bn|+Am+1δ∑n=m+1∞‍|an|≤1.
From condition (50), it is sufficient to show that
(60)∑n=2∞‍|an|+∑n=2∞‍|bn|+Am+1δ∑n=m+1∞‍|an|≤∑n=2∞‍Anδ|an|+∑n=2∞‍Bnδ|an|
which is equivalently to
(61)∑n=2m‍(An-δδ)|an|+∑n=2∞‍(Bn-δδ)|bn|+∑n=m+1∞‍(An-An+1δ)|an|≥0.
To see that the function given by (55) gives the sharp result, we observe that for z=reiπ/n(62)f(z)fm(z)=1+δAm+1zm⟶1-δAm+1=Am+1-δAm+1whenr⟶1-.We next determine bounds forℜ{fm(z)/f(z)}.Theorem 14.
Iff of the form (6) with b1=0 satisfies condition (50), then
(63)ℜ{fm(z)f(z)}≥Am+1Am+1+δ,(z∈𝒰),
where
(64)An≥{δ,ifn=2,3,…,m,Am+1,ifn=m+1,m+2,…,Bn≥δ,ifn=2,3,….
The result (63) is sharp with the function given by
(65)f(z)=z+δAm+1zm+1.Proof.
Define the functionw(z) by
(66)1+w(z)1-w(z)=Am+1+δδ[fm(reiθ)f(reiθ)-Am+1Am+1+δ]=((∑n=m+1∞‍anrn-1ei(n-1)θ)1+∑n=2m‍anrn-1ei(n-1)θ0000+∑n=2∞‍bn¯rn-1e-i(n+1)θ0000-Am+1δ(∑n=m+1∞‍anrn-1ei(n-1)θ))×(1+∑n=2m‍anrn-1ei(n-1)θ000000+∑n=2∞‍bn¯rn-1e-i(n+1)θ)-1.
Hence we obtain
(67)|w(z)|≤(Am+1+δδ(∑n=m+1∞|an|))×(2-2[∑n=2m|an|+∑n=2∞|bn|]000000-((Am+1-δ)/δ)∑n=m+1∞|an|)-1≤1.
The last inequality is equivalent to
(68)∑n=2ℓ‍|an|+∑n=2∞‍|bn|+Am+1δ∑n=m+1∞‍|an|≤1.
Making use of (50) and the condition (64), we obtain (61). Finally, equality holds in (63) for the extremal function f(z) given by (65).We next turns to ratios forℜ{f′(z)/fm′(z)} and ℜ{fm′(z)/f′(z)}.Theorem 15.
Iff of the form (6) with b1=0 satisfies condition (50), then
(69)ℜ{f′(z)fm′(z)}≥Am+1-(m+1)δAm+1,(z∈𝒰),
where
(70)An≥{δ,ifn=2,3,…,m,Am+1,ifn=m+1,m+2,…,Bn≥δ,ifn=2,3,….
The result (69) is sharp with the function given by f(z)=z+(δ/Am+1)zm+1.Proof.
Define the functionw(z) by
(71)1+w(z)1-w(z)=Am+1(m+1)δ[f′(z)fm′(z)-Am+1-(m+1)δAm+1]=((∑n=m+1∞‍nanrn-1ei(n-1)θ)1+∑n=2m‍nanrn-1ei(n-1)θ0000+∑n=2∞‍nbn¯rn-1e-i(n+1)θ0000-Am+1(m+1)δ(∑n=m+1∞‍nanrn-1ei(n-1)θ))×(1+∑n=2m‍nanrn-1ei(n-1)θ000000-∑n=2∞‍nbn¯rn-1e-i(n+1)θ)-1.
The result (69) follows by using the techniques as used in Theorem 13.Proceeding exactly as in the proof of Theorem14, we can prove the following theorem.Theorem 16.
Iff of the form (6) with b1=0 satisfies the condition (50), then
(72)ℜ{fm′(z)f′(z)}≥Am+1Am+1+(m+1)δ,(z∈𝒰).
The result is sharp with the function given by f(z)=z+(δ/Am+1)zm+1.We next determine bounds forℜ{f(z)/fk(z)} and ℜ{fk(z)/f(z)}.Theorem 17.
Iff of the form (6) with b1=0 satisfies condition (50), then
(73)ℜ{f(z)fk(z)}≥Bk+1-δBk+1,(z∈𝒰),
where
(74)Bn≥{δ,ifn=2,3,…,k,Bk+1,ifn=k+1,k+2,…,An≥δ,ifn=2,3,….
The result (73) is sharp with the function given by f(z)=z+(δ/Bk+1)z¯k+1.Theorem 18.
Iff of the form (6) with b1=0 satisfies condition (50), then
(75)ℜ{fk(z)f(z)}≥Bk+1Bk+1+δ,(z∈𝒰),
where
(76)Bn≥{δ,ifn=2,3,…,k,Bk+1,ifn=k+1,k+2,…,An≥δ,ifn=2,3,….
The result (75) is sharp with the function given by f(z)=z+(δ/Bk+1)z¯k+1.Proof.
Define the functionw(z) by
(77)1+w(z)1-w(z)=Bk+1+δδ[fk(reiθ)f(reiθ)-Bk+1Bk+1+δ]=(1+∑n=2∞‍anrn-1ei(n-1)θ+∑n=2k‍bn¯rn-1e-i(n+1)θ00000-Bk+1Bk+1+δ∑n=2k+1‍bn¯rn-1e-i(n-1)θ)×(∑n=2k‍bn¯rn-1e-i(n+1)θ1+∑n=2∞‍anrn-1ei(n-1)θ000000+∑n=2k‍bn¯rn-1e-i(n+1)θ)-1.
We omit the details of proof, because it runs parallel to that from Theorem 14.Theorem 19.
Iff of the form (6) with b1=0 satisfies condition (50), then
(78)ℜ{f(z)fm,k(z)}≥Am+1-δAm+1,(z∈𝒰),
where
(79)An≥{δ,ifn=2,3,…,m,m+1,Am+1,ifn=m+1,m+2,…,Bn≥{δ,ifn=2,3,…,m,Am+1,ifn=m+1,m+2,….
The result (78) is sharp with the function given by f(z)=z+(δ/Am+1)zm+1.Theorem 20.
Iff of the form (6) with b1=0 satisfies condition (50), then
(80)ℜ{f(z)fm,k(z)}≥Bk+1-δBk+1,(z∈𝒰)
where
(81)Bn≥{δ,ifn=2,3,…,k,Bk+1,ifn=k+1,k+2,…,An≥{δ,ifn=2,3,…,k,Bk+1,ifn=k+1,k+2,….Theorem 21.
Iff of the form (6) with b1=0 satisfies condition (50), then
(82)ℜ{fm,k(z)f(z)}≥Am+1Am+1+δ,(z∈𝒰).Theorem 22.
Iff of the form (6) with b1=0 satisfies condition (50), then
(83)ℜ{fm,k(z)f(z)}≥Bk+1Bk+1+δ,(z∈𝒰).
The result (83) is sharp with the function given by f(z)=z+(δ/Bk+1)z¯k+1.Theorem 23.
Iff of the form (6) with b1=0 satisfies condition (50), then
(84)ℜ{f′(z)fm′(z)}≥Am+1-(m+1)δAm+1,(z∈𝒰),
where
(85)An≥{δ,ifn=2,3,…,m,Am+1,ifn=m+1,m+2,…,Bn≥{δ,ifn=2,3,…,m,Am+1,ifn=m+1,m+2,….
The result (84) is sharp with the function given by f(z)=z+(δ/Am+1)zm+1.Theorem 24.
Iff of the form (6) with b1=0 satisfies condition (50), then
(86)ℜ{fm,k′(z)f′(z)}≥Am+1Am+1+(m+1)δ,(z∈𝒰).
The result (86) is sharp with the function given by f(z)=z+(δ/Am+1)zm+1.Concluding Remarks. By choosing λ=0 (or λ=1) and ℓ=0 (or ℓ=1), the various results presented in this paper would provide interesting extensions and generalizations of the subclasses of harmonic star-like functions with positive coefficients of order γ(1<γ≤4/3) based on Salagean operator and similarly for convex functions. The details involved in the derivations of such specializations of the results presented in this paper are fairly straight-forward and hence omitted.



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*Source: 793709-2014-04-06.xml*