The purpose of the present paper is to establish some new results giving the sharp bounds of the real parts of ratios of harmonic univalent functions to their sequences of partial sums by using convolution. Relevant connections of the results presented here with various known results are briefly indicated.

1. Introduction

A continuous complex-valued function f=u+iv is said to be harmonic in a simply connected domain D if both u and v are real harmonic in D. In any simply-connected domain we can write f=h+g¯, where h and g are analytic in D. We call h the analytic part and g the co-analytic part of f. A necessary and sufficient condition for f to be locally univalent and sense-preserving in D is that |h'(z)|>|g'(z)|,z∈D, see [1]. For more basic results on harmonic functions one may refer to the following standard text book by Duren [2]. See also Ahuja [3] and Ponnusamy and Rasila ([4, 5]).

Denote by SH the class of functions f=h+g¯ which are harmonic univalent and sense-preserving in the open unit disk U={z:|z|<1} for which f(0)=fz(0)-1=0. Then for f=h+g¯∈SH we may express the analytic functions h and g as
(1.1)h(z)=z+∑k=2∞akzk,g(z)=∑k=1∞bkzk,|b1|<1.

Note that SH reduces to the class S of normalized analytic univalent functions, if the coanalytic part of its member is zero, that is, g≡0, and for this class f(z) may be expressed as
(1.2)f(z)=z+∑k=2∞akzk.

Let ϕ(z)∈SH be a fixed function of the form
(1.3)ϕ(z)=z+∑k=2∞ckzk+∑k=1∞dkzk¯,(dk≥ck≥c2>0;k≥2,|d1|<1).

Now, we introduce a class SH(ck,dk,δ) consisting of functions of the form (1.1) which satisfies the inequality
(1.4)∑k=2∞ck|ak|+∑k=1∞dk|bk|≤δ,whereδ>0,
and we note that if dk=0, then the class SH(ck,dk,δ) reduces to the class Sϕ(ck,δ) which was introduced by Frasin [6]. In this case the condition (1.4) reduces to
(1.5)∑k=2∞ck|ak|≤δ,whereδ>0.

It is easy to see that various subclasses of SH consisting of functions f(z) of the form (1.1) can be represented as SH(ck,dk,δ) for suitable choices of ck,dk, and δ studied earlier by various researchers. For example:

SH(k,k,1)≡SH* and SH(k2,k2,1)≡KH studied by Silverman [7]; Silverman and Silvia [8].

SH(k-α,k+α,1-α)≡SH*(α) studied by Jahangiri [9].

SH((k-α)(ϕ(k,λ)),(k+α)(ϕ(k,λ)),1-α)≡SH,λ*(α) studied by Dixit and Porwal [10].

SH(km-αkn,km-αkn,1-α)≡HS(m,n,α) studied by Dixit and Porwal [11].

SH(k,k,β-1)≡HP(β) studied by Dixit and Porwal [12].

SH(λk(1-αλ)-α(1-λ),μk(1-αλ)+α(1-λ),1-α)≡SH(Φ,Ψ,α,λ) studied by Dixit and Porwal [13].

SH(λk-α,μk+α,1-α)≡SH(Φ,Ψ,α) studied by Frasin [14].

SH(k(1-αλ)-α(1-λ),k(1-αλ)+α(1-λ),1-α)≡SH*(α,λ) studied by Öztürk et al. [15].

SH((k(β+1)-t(β+γ))Γ(α1,k),(k(β+1)+t(β+γ))Γ(α1,k)),(k-1)!(1-γ))≡GH(α1,β,γ,t) studied by Porwal et al. [16].

SH(2k-1-α,2k+1+α,1-α)≡GH(α) studied by Rosy et al. [17].

In 1985, Silvia [18] studied the partial sums of convex functions of order α. Later on, Silverman [19], Afaf et al. [20], Dixit and Porwal [21], Frasin ([6, 22]), Murugusundaramoorthy et al. [23], Owa et al. [24], Porwal and Dixit [25], Raina and Bansal [26] and Rosy et al. [27] studied and generalized the results on partial sums for various classes of analytic functions. Very recently, Porwal [28], Porwal and Dixit [29] studied analogues interesting results on the partial sums of certain harmonic univalent functions. In this work, we extend all these results.

Now, we let the sequences of partial sums of function of the form (1.1) with b1=0 be
(1.6)fm(z)=z+∑k=2makzk+∑k=2∞bkzk¯,fn(z)=z+∑k=2∞akzk+∑k=2nbkzk¯,fm,n(z)=z+∑k=2makzk+∑k=2nbkzk¯,
when the coefficients of f are sufficiently small to satisfy the condition (1.4).

In the present paper, we determine sharp lower bounds for Re{(f(z)*ψ(z))/(fm(z)*ψ(z))}, Re{(fm(z)*ψ(z))/(f(z)*ψ(z))}, Re{(f(z)*ψ(z))/(fn(z)*ψ(z))}, Re{(fn(z)*ψ(z))/(f(z)*ψ(z))}, Re{(f(z)*ψ(z))/(fm,n(z)*ψ(z))}, and Re{(fm,n(z)*ψ(z))/(f(z)*ψ(z))} where fm(z), fn(z) and fm,n(z) are defined above and ψ(z)=z+∑k=2∞λkzk+∑k=2∞μkzk¯, (λk≥0,μk≥0) is a harmonic function and the operator “*” stands for the Hadamard product or convolution of two power series, which is defined for two functions f(z) and g(z) are of the form
(1.7)f(z)=z+∑k=2∞akzk+∑k=2∞bkzk¯,g(z)=z+∑k=2∞ckzk+∑k=2∞dkzk¯,
as
(1.8)(f*g)(z)=f(z)*g(z)=z+∑k=2∞akckzk+∑k=2∞bkdkzk¯.
It is worthy to note that this study not only gives as a particular case, the results of Porwal [28], Porwal and Dixit [29], but also give rise to several new results.

2. Main Results

In our first theorem, we determine sharp lower bounds for Re{(f(z)*ψ(z))/(fm(z)*ψ(z))}.

Theorem 2.1.

If f of the form (1.1) with b1=0, satisfies the condition (1.4), then
(2.1)Re{f(z)*ψ(z)fm(z)*ψ(z)}≥cm+1-λm+1δcm+1,(z∈U),
where
(2.2)ck≥{λkδifk=2,3,…,m,λkcm+1λm+1ifk=m+1,m+2….

The result (2.1) is sharp with the function given by
(2.3)f(z)=z+δcm+1zm+1,
where 0<δ≤cm+1/λm+1.

Proof.

To obtain sharp lower bound given by (2.1), let us put
(2.4)1+ω(z)1-ω(z)=cm+1λm+1δ[f(reiθ)*ψ(reiθ)fm(reiθ)*ψ(reiθ)-cm+1-λm+1δcm+1]=1+∑k=2mλkakrk-1ei(k-1)θ+∑k=2∞μkbk¯rk-1e-i(k+1)θ+(cm+1/λm+1δ)[∑k=m+1∞λkakrk-1ei(k-1)θ]1+∑k=2mλkakrk-1ei(k-1)θ+∑k=2∞μkbk¯rk-1e-i(k+1)θ.
So that
(2.5)ω(z)=ℛ[∑k=m+1∞λkakrk-1ei(k-1)θ]2+2(∑k=2mλkakrk-1ei(k-1)θ+∑k=2∞μkb-krk-1e-i(k+1)θ)+ℛ(∑k=m+1∞λkakrk-1ei(k-1)θ),
where ℛ denotes (cm+1/λm+1δ).

Then
(2.6)|ω(z)|≤(cm+1/λm+1δ)[∑k=m+1∞λk|ak|]2-2(∑k=2mλk|ak|+∑k=2∞μk|bk|)-(cm+1/λm+1δ)(∑k=m+1∞λk|ak|).
This last expression is bounded above by 1, if and only if
(2.7)∑k=2mλk|ak|+∑k=2∞μk|bk|+cm+1λm+1δ(∑k=m+1∞λk|ak|)≤1.
It suffices to show that L.H.S. of (2.7) is bounded above by ∑k=2∞(ck/δ)|ak|+∑k=2∞(dk/δ)|bk|, which is equivalent to
(2.8)∑k=2mck-λkδδ|ak|+∑k=2∞dk-μkδδ|bk|+∑k=m+1∞ckλm+1-cm+1λkλm+1δ|ak|≥0.

To see that f(z)=z+(δ/cm+1)zm+1 gives the sharp result, we observe that for z=reiπ/m that
(2.9)f(z)*ψ(z)fm(z)*ψ(z)=1+λm+1δcm+1zm→1-λm+1δcm+1=cm+1-λm+1δcm+1,
when r→1-.

We next determine bounds for Re{(fm(z)*ψ(z))/(f(z)*ψ(z))}.

Theorem 2.2.

If f of the form (1.1) with b1=0, satisfies the condition (1.4), then
(2.10)Re{fm(z)*ψ(z)f(z)*ψ(z)}≥cm+1cm+1+λm+1δ,(z∈U),
where
(2.11)ck≥{λkδifk=2,3,…,mλkcm+1λm+1ifk=m+1,m+2….
The result (2.10) is sharp with the function given by (2.3).

Proof.

To prove Theorem 2.2, we may write
(2.12)1+ω(z)1-ω(z)=cm+1+λm+1δλm+1δ[fm(z)*ψ(z)f(z)*ψ(z)-cm+1cm+1+λm+1δ]=1+∑k=2mλkakrk-1ei(k-1)θ+∑k=2∞μkbk¯rk-1e-i(k+1)θ-(cm+1/λm+1δ)[∑k=m+1∞λkakrk-1ei(k-1)θ]1+∑k=2∞λkakrk-1ei(k-1)θ+∑k=2∞μkbk¯rk-1e-i(k+1)θ,
where
(2.13)|ω(z)|≤((cm+1+λm+1δ)/λm+1δ)[∑k=m+1∞λk|ak|]2-2(∑k=2mλk|ak|+∑k=2∞μk|bk|)-((cm+1-λm+1δ)/λm+1δ)(∑k=m+1∞λk|ak|)≤1.
This last inequality is equivalent to
(2.14)∑k=2mλk|ak|+∑k=2∞μk|bk|+cm+1λm+1δ(∑k=m+1∞λk|ak|)≤1.
Since the L.H.S. of (2.14) is bounded above by ∑k=2∞(ck/δ)|ak|+∑k=2∞(dk/δ)|bk|, the proof is evidently complete.

Adopting the same procedure as in Theorems 2.1 and 2.2 and performing simple calculations, we can obtain the sharp lower bounds for the real parts of the following ratios:
(2.15)Re{(f(z)*ψ(z))(fn(z)*ψ(z))},Re{(fn(z)*ψ(z))(f(z)*ψ(z))},Re{(f(z)*ψ(z))(fm,n(z)*ψ(z))},Re{(fm,n(z)*ψ(z))(f(z)*ψ(z))}.

The results corresponding to real parts of these ratios are contained in the following Theorems 2.3, 2.4, 2.5, and 2.6.

Theorem 2.3.

If f of the form (1.1) with b1=0 satisfies the condition (1.4), then
(2.16)Re{f(z)*ψ(z)fn(z)*ψ(z)}≥dn+1-μn+1δdn+1,(z∈U),
where
(2.17)dk≥{μkδifk=2,3,…,n,μkdn+1μn+1ifk=n+1,n+2….

The result (2.16) is sharp with the function
(2.18)f(z)=z+δdn+1z-n+1.

Theorem 2.4.

If f of the form (1.1) with b1=0, satisfies the condition (1.4), then
(2.19)Re{fn(z)*ψ(z)f(z)*ψ(z)}≥dn+1dn+1+μn+1δ,(z∈U),
where
(2.20)dk≥{μkδifk=2,3,…,n,μkdn+1μn+1ifk=n+1,n+2….

The result (2.19) is sharp with the function given by (2.18).

Theorem 2.5.

If f of the form (1.1) with b1=0, satisfies the condition (1.4), then

where
(2.28)ck≥{μkδifk=2,3,…,m,μkdn+1μn+1ifk=m+1,m+2…,dk≥{μkδifk=2,3,…,n,μkdn+1μn+1ifk=n+1,n+2….

The results (2.25) and (2.27) are sharp with the functions given by (2.3) and (2.18) respectively.

3. Some Consequences and Concluding Remarks

In this section, we specifically point out the relevances of some of our main results with those results which have appeared recently in literature.

If we put ψ(z)=(z/(1-z))+((z/(1-z))-z)¯ and ψ(z)=(z/(1-z)2)+((z/(1-z)2)-z¯) in Theorems 2.1–2.6, then we obtain the corresponding results of Porwal [28].

Next, if we put ψ(z)=(z/(1-z))+((z/(1-z))-z)¯, ψ(z)=(z/(1-z)2)+((z/(1-z)2)-z)¯, ck=k-α,dk=k+α, and δ=1-α in Theorems 2.1–2.6, then we obtain the corresponding results of Porwal and Dixit [29].

Again, if we put g=0 in Theorems 2.1 and 2.2, then we obtain the corresponding results of Dixit and Porwal [21].

Lastly, if we put g≡0, ψ(z)=z/(1-z), and ψ(z)=z/(1-z)2 Theorems 2.1 and 2.2, then we obtain the result of Frasin [6].

We mention below some corollaries giving sharp bounds of the real parts on the ratio of univalent functions to its sequences of partial sums.

By putting ψ(z)=z/(1-z) in Theorem 2.1 for the function f of the form (1.2) with ck=k-α and δ=1-α, then we obtain the following result of Silverman [19], Theorem 1.

Corollary 3.1.

If f of the form (1.2) satisfies the condition (1.5) with ck=k-α and δ=1-α, then
(3.1)Re{f(z)fm(z)}≥mm+1-α,z∈U.
The result is sharp for every m, with the extremal function given by
(3.2)f(z)=z+1-αm+1-αzm+1.

On the other hand, if we put ψ(z)=z/(1-z)2 in Theorem 2.1 for the function f of the form (1.2) with ck=k-α and δ=1-α, then we obtain the following result of Silverman, Theorem 4(i) [19].

Corollary 3.2.

If f of the form (1.2) satisfies the condition (1.5) with ck=k-α, then for z∈U(3.3)Re{f′(z)fm′(z)}≥αmm+1-α.
The result is sharp for every m, with the extremal function given by (3.2).

Also, if we put ψ(z)=z/(1-z) in Theorem 2.1 for the function f of the form (1.2) belonging to the class Sϕ(ck,δ), then we obtain the following result of Frasin [6].

Corollary 3.3.

If ∈Sϕ(ck,δ), then
(3.4)Re{f(z)fm(z)}≥cm+1-δcm+1,(z∈U),
where
(3.5)ck≥{δifk=2,3,…,m,cm+1ifk=m+1,m+2,….

The result is sharp for every m, with the extremal function given by
(3.6)f(z)=z+δcm+1zm+1.

Next, if we put ψ(z)=z/(1-z) in Theorem 2.1 for the function f of the form (1.2) with ck=ρk(λ,γ,η) and δ=1-γ, then we obtain the following result of Murugusundaramoorthy et al. ([23], Theorem 2.1).

Corollary 3.4.

If f of the form (1.2) satisfies the condition (1.5) with ck=ρk(λ,γ,η) and δ=1-γ, then for z∈U:(3.7)Re{f(z)fm(z)}≥ρm+1(λ,γ,η)-1+γρm+1(λ,γ,η),
where
(3.8)ρk(λ,γ,η)≥{1-γifk=2,3,…,m,ρm+1(λ,γ,η)ifk=m+1,m+2….

The result is sharp for every m, with the extremal function given by
(3.9)f(z)=z+1-γρm+1(λ,γ,η)zm+1.

Again, if we set ψ(z)=(z/(1-z))+((z/(1-z))-z¯) in Theorem 2.1, then we obtain the following result of Porwal [28].

Corollary 3.5.

If f of the form (1.1) with b1=0, satisfies the condition (1.4) with
(3.10)ck≥{δifk=2,3,…,m,cm+1ifk=m+1,m+2…,
then
(3.11)Re{f(z)fm(z)}≥cm+1-δcm+1,∀z∈U.
The result (3.11) is sharp with the function given by (3.6).

Here we give some open problems for the readers.

In 2004, Owa et al. [24] studied the starlikeness and convexity properties on the partial sums fn(z) and gn(z) of the familiar Koebe function f(z)=z/(1-z)2 which is the extremal function for the class S* of starlike functions in the open unit disk U and the function g(z)=z/(1-z) which is the extremal function for the class K of convex functions in the open unit disk U, respectively. They also presented some illustrative examples by using Mathematica (Version 4.0). It is interesting to obtain analogues results on harmonic starlikeness and convexity properties of the partial sums of the harmonic Koebe function.

In 2003, Jahangiri et al. [30] studied the construction of sense-preserving, univalent, and close-to-convex harmonic functions by using of the Alexander integral transforms of certain analytic functions (which are starlike or convex of positive order). They construct a function
(3.12)g(z)=z+1-αk-αzk+α2(z¯)2+α(1-α)(k+1)(k-α)(z¯)k+1,
which is sense-preserving, univalent, and close-to-convex harmonic in U, by using the result of Theorem 2 [30] and taking the following function:
(3.13)f(z)=z+1-αk(k-α)zk,(k>1;0≤α<1).
It is worthy to note that the function (3.13) is of the form (3.6) with ck=k(k-α) and δ=1-α. Therefore, it is natural to ask that the results of [30] may be generalized for the function of the form (3.6).

Acknowledgment

The author is thankful to the referee for his valuable comments and observations which helped in improving the paper.

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