A Subclass of Harmonic Univalent Functions with Varying Arguments Defined by Generalized Derivative Operator

Abstract

Making use of the generalized derivative operator, we introduce a new class of complex valued harmonic functions which are orientation preserving and univalent in the open unit disc and are related to uniformly convex functions. We investigate the coefficient bounds, neighborhood, and extreme points for this generalized class of functions.

1. Introduction

A continuous complex-valued function 𝑓=𝑢+𝑖𝑣 defined in a simply connected complex domain 𝐷 is said to be harmonic in 𝐷 if both 𝑢 and 𝑣 are real harmonic in 𝐷. Such functions can be expressed as 𝑓=ℎ+𝑔, (1.1) 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 sense preserving in 𝐷 is that |ℎ(𝑧)|>|𝑔(𝑧)| for all 𝑧 in 𝐷 (see [1]). Let 𝐻 be the class of functions of the form (1.1) that are harmonic univalent and sense preserving in the unit disk 𝑈={𝑧∶|𝑧|<1} for which 𝑓(0)=𝑓_𝑧(0)−1=0. Then for 𝑓=ℎ+𝑔∈𝐻, we may express the analytic functions ℎ and 𝑔 asℎ(𝑧)=𝑧+^∞_𝑘=2𝑎_𝑘𝑧^𝑘,𝑔(𝑧)=𝑧+^∞_𝑘=1𝑏_𝑘𝑧^𝑘,𝑧∈𝑈,𝑏₁<1. (1.2) In 1984, Clunie and Sheil-Small [1] investigated the class 𝑆_𝐻 as well as its geometric subclasses and obtained some coefficient bounds. Since then, there have been several related papers on 𝑆_𝐻 and its subclasses. Now we will introduce a generalized derivative operator for 𝑓=ℎ+𝑔 given by (1.2). For fixed positive natural 𝑚 and 𝜆₂≥𝜆₁≥0,𝐷_𝜆^𝑚,𝑘_1,𝜆2𝑓(𝑧)=𝐷_𝜆^𝑚,𝑘_1,𝜆2ℎ(𝑧)+𝐷_𝜆^𝑚,𝑘_1,𝜆2𝑔(𝑧),𝑧∈𝑈, (1.3) where𝐷_𝜆^𝑚,𝑘_1,𝜆2ℎ(𝑧)=𝑧+^∞_𝑘=21+𝜆₁+𝜆₂(𝑘−1)1+𝜆₂(𝑘−1)^𝑚𝑎_𝑘𝑧^𝑘,𝐷_𝜆^𝑚,𝑘_1,𝜆2𝑔(𝑧)=^∞_𝑘=11+𝜆₁+𝜆₂(𝑘−1)1+𝜆₂(𝑘−1)^𝑚𝑎_𝑘𝑧^𝑘. (1.4) We note that by specializing the parameters, especially when 𝜆₁=𝜆₂=0, 𝐷_𝜆^𝑚,𝑘_1,𝜆2 reduces to 𝐷^𝑚 which is introduced by Salagean in [2].

Now we will introduce the following definition.

Definition 1.1. For 0≤ℓ<1, let 𝐺_𝐻(ℓ,𝑚,𝑘,𝜆₁,𝜆₂) denote the subfamily of starlike harmonic functions 𝑓∈𝐻 of the form (1.1) such that Re1+𝑒^𝑖𝜓𝑧𝐷_𝜆^𝑚,𝑘_1,𝜆2𝑓(𝑧)^′𝑧^′𝐷_𝜆^𝑚,𝑘_1,𝜆2𝑓(𝑧)−𝑒^𝑖𝜓≥ℓ (1.5) for a suitable real 𝜓 and 𝑧∈𝑈 where (𝐷_𝜆^𝑚,𝑘_1,𝜆2𝑓(𝑧))^′=(𝑑/𝑑𝜃)(𝐷_𝜆^𝑚,𝑘_1,𝜆2𝑓(𝑟𝑒^𝑖𝜃)),(𝑑/𝑑𝜃)(𝑧=𝑟𝑒^𝑖𝜃).
We also let 𝑉_𝐻(ℓ,𝑚𝑘,𝜆₁,𝜆₂)=𝐺_𝐻(ℓ,𝑚𝑘,𝜆₁,𝜆₂)∩𝑉_𝐻 where 𝑉_𝐻 is the class of harmonic functions with varying arguments introduced by Jahangiri and Silverman [3] consisting of functions 𝑓 of the form (1.1) in 𝐻 for which there exists a real number 𝜙 such that 𝜂_𝑘+(𝑘−1)𝜙≡𝜋(mod2𝜋),𝛿_𝑘+(𝑘−1)𝜙≡0(𝑘≥2), (1.6) where 𝜂_𝑘=arg(𝑎_𝑘) and 𝛿_𝑘=arg(𝑏_𝑘). The same class introduced in [4] with different differential operator.
In this paper, we obtain a sufficient coefficient condition for functions 𝑓 given by (1.2) to be in the class 𝐺_𝐻(ℓ,𝑚,𝑘,𝜆₁,𝜆₂). It is shown that this coefficient condition is necessary also for functions belonging to the class 𝑉_𝐻(ℓ,𝑚,𝑘,𝜆₁,𝜆₂). Further, extreme points for functions in 𝑉_𝐻(ℓ,𝑚,𝑘,𝜆₁,𝜆₂) are also obtained.

2. Main Result

We begin deriving a sufficient coefficient condition for the functions belonging to the class 𝐺_𝐻(ℓ,𝑚,𝑘,𝜆₁,𝜆₂). This result is contained in the following.

Theorem 2.1. Let 𝑓=ℎ+𝑔 given by (1.2). Furthermore, let ^∞_𝑘=22𝑘−1−ℓ𝑎1−ℓ^𝑘+2𝑘+1+ℓ𝑏1−ℓ^𝑘1+𝜆₁+𝜆₂(𝑘−1)1+𝜆₂(𝑘−1)^𝑚≤1−3+ℓ𝑏3−ℓ¹, (2.1) where 0≤ℓ<1, then 𝑓∈𝐺_𝐻(ℓ,𝑚,𝑘,𝜆₁,𝜆₂).

Proof. We first show that if the inequality (2.1) holds for the coefficients of 𝑓=ℎ+𝑔, then the required condition (1.5) is satisfied. Using (1.3) and (1.5), we can write Re1+𝑒^𝑖𝜓𝑧𝐷_𝜆^𝑚,𝑘_1,𝜆2ℎ(𝑧)^′−𝑧𝐷_𝜆^𝑚,𝑘_1,𝜆2𝑔(𝑧)^′𝐷_𝜆^𝑚,𝑘_1,𝜆2ℎ(𝑧)+𝐷_𝜆^𝑚,𝑘_1,𝜆2𝑔(𝑧)−𝑒^𝑖𝜓𝐴(𝑧)=Re𝐵(𝑧), (2.2) where 𝐴(𝑧)=1+𝑒^𝑖𝜓𝑧𝐷_𝜆^𝑚,𝑘_1,𝜆2ℎ(𝑧)^′−𝑧𝐷_𝜆^𝑚,𝑘_1,𝜆2𝑔(𝑧)^′−𝑒^𝑖𝜓𝐷_𝜆^𝑚,𝑘_1,𝜆2ℎ(𝑧)+𝐷_𝜆^𝑚,𝑘_1,𝜆2𝑔(𝑧),𝐵(𝑧)=𝐷_𝜆^𝑚,𝑘_1,𝜆2ℎ(𝑧)+𝐷_𝜆^𝑚,𝑘_1,𝜆2𝑔(𝑧). (2.3)
In view of the simple assertion that Re(𝑤)≥ℓ if and only if |1−ℓ+𝑤|≥|1+ℓ−𝑤|, it is sufficies to show that |𝐴(𝑧)+(1−ℓ)𝐵(𝑧)|−|𝐴(𝑧)−(1+ℓ)𝐵(𝑧)|≥0. (2.4)
Substituting for 𝐴(𝑧) and 𝐵(𝑧) the appropriate expressions in (2.4), we get |𝐴(𝑧)+()𝐵(𝑧−|𝐴(𝑧)−()𝐵(𝑧≥(𝑧|−1−ℓ)|1+ℓ)|2−ℓ)|^∞_𝑘=2()𝜆2𝑘−ℓ1+₁+𝜆₂(𝑘−1)1+𝜆₂(𝑘−1)^𝑚𝑎_𝑘‖𝑧^𝑘−^∞_𝑘=2()𝜆2𝑘+ℓ1+₁+𝜆₂(𝑘−1)1+𝜆₂(𝑘−1)^𝑚𝑏_𝑘‖𝑧^𝑘−ℓ|𝑧|−^∞_𝑘=2()𝜆2𝑘−2−ℓ1+₁+𝜆₂(𝑘−1)1+𝜆₂(𝑘−1)^𝑚𝑎_𝑘‖𝑧^𝑘−^∞_𝑘=2()𝜆2𝑘+2+ℓ1+₁+𝜆₂(𝑘−1)1+𝜆₂(𝑘−1)^𝑚𝑏_𝑘‖𝑧^𝑘≥2(1−ℓ)|𝑧|1−3+ℓ𝑏1−ℓ¹−^∞_𝑘=22𝑘−1−ℓ𝜆1−ℓ1+₁+𝜆₂(𝑘−1)1+𝜆₂(𝑘−1)^𝑚𝑎_𝑘+2𝑘+1+ℓ𝜆1−ℓ1+₁+𝜆₂(𝑘−1)1+𝜆₂(𝑘−1)^𝑚𝑏_𝑘≥0 (2.5) by virtue of the inequality (2.1). This implies that 𝑓∈𝐺_𝐻(ℓ,𝑚,𝑘,𝜆₁,𝜆₂).
Now we obtain the necessary and sufficient condition for function 𝑓=ℎ+𝑔 be given with condition (1.6).

Theorem 2.2. Let 𝑓=ℎ+𝑔 be given by (1.2). Then 𝑓∈𝑉_𝐻(ℓ,𝑚,𝑘,𝜆₁,𝜆₂) if and only if ^∞_𝑘=22𝑘−1−ℓ𝑎1−ℓ^𝑘+2𝑘+1+ℓ𝑏1−ℓ^𝑘1+𝜆₁+𝜆₂(𝑘−1)1+𝜆₂(𝑘−1)^𝑚≤1−3+ℓ𝑏3−ℓ¹, (2.6) where 0≤ℓ<1.

Proof. Since 𝑉_𝐻(ℓ,𝑚,𝑘,𝜆₁,𝜆₂)⊂𝐺_𝐻(ℓ,𝑚,𝑘,𝜆₁,𝜆₂), we only need to prove the necessary part of the theorem. Assume that 𝑓∈𝑉_𝐻(ℓ,𝑚,𝑘,𝜆₁,𝜆₂), then by virtue of (1.3) to (1.5), we obtain Re1+𝑒^𝑖𝜓𝑧𝐷_𝜆^𝑚,𝑘_1,𝜆2ℎ(𝑧)^′−𝑧𝐷_𝜆^𝑚,𝑘_1,𝜆2𝑔(𝑧)^′𝐷_𝜆^𝑚,𝑘_1,𝜆2ℎ(𝑧)+𝐷_𝜆^𝑚,𝑘_1,𝜆2𝑔(𝑧)−𝑒^𝑖𝜓+ℓ≥0. (2.7)
The above inequality is equivalent to Re𝑧+^∞_𝑘=2𝑘1+𝑒^𝑖𝜓−ℓ−𝑒^𝑖𝜓1+𝜆₁+𝜆₂(𝑘−1)1+𝜆₂(𝑘−1)^𝑚𝑎_𝑘𝑧^𝑘−^∞_𝑘=2𝑘1+𝑒^𝑖𝜓+ℓ+𝑒^𝑖𝜓1+𝜆₁+𝜆₂(𝑘−1)1+𝜆₂(𝑘−1)^𝑚𝑏_𝑘𝑧^𝑘,×𝑧+^∞_𝑘=21+𝜆₁+𝜆₂(𝑘−1)1+𝜆₂(𝑘−1)^𝑚𝑎_𝑘𝑧^𝑘+^∞_𝑘=21+𝜆₁+𝜆₂(𝑘−1)1+𝜆₂(𝑘−1)^𝑚𝑏_𝑘𝑧^𝑘−1=Re(1−ℓ)+^∞_𝑘=2𝑘1+𝑒^𝑖𝜓−ℓ−𝑒^𝑖𝜓1+𝜆₁+𝜆₂(𝑘−1)1+𝜆₂(𝑘−1)^𝑚𝑎_𝑘𝑧^𝑘−1−𝑧𝑧^∞𝑘=2𝑘1+𝑒^𝑖𝜓+ℓ+𝑒^𝑖𝜓1+𝜆₁+𝜆₂(𝑘−1)1+𝜆₂(𝑘−1)^𝑚𝑏_𝑘𝑧^𝑘−1×1+^∞_𝑘=21+𝜆₁+𝜆₂(𝑘−1)1+𝜆₂(𝑘−1)^𝑚𝑎_𝑘𝑧^𝑘−1+𝑧𝑧^∞𝑘=21+𝜆₁+𝜆₂(𝑘−1)1+𝜆₂(𝑘−1)^𝑚𝑏_𝑘𝑧^𝑘−1−1≥0. (2.8)
This condition must hold for all values of 𝑧, such that |𝑧|=𝑟<1. Upon choosing 𝜙 according to (1.6) and noting that Re(−𝑒^𝑖𝜓)≥−|𝑒^𝑖𝜓|=−1, the above inequality reduces to (1−ℓ)−1−𝑏₁−^∞_𝑘=2()𝜆2𝑘−1−ℓ1+₁+𝜆₂(𝑘−1)1+𝜆₂(𝑘−1)^𝑚𝑎_𝑘𝑟^𝑘−1+()𝜆2𝑘+1+ℓ1+₁+𝜆₂(𝑘−1)1+𝜆₂(𝑘−1)^𝑚𝑏_𝑘𝑟^𝑘−1×1−^∞_𝑘=21+𝜆₁+𝜆₂(𝑘−1)1+𝜆₂(𝑘−1)^𝑚𝑎_𝑘𝑟^𝑘−1+^∞_𝑘=11+𝜆₁+𝜆₂(𝑘−1)1+𝜆₂(𝑘−1)^𝑚𝑏_𝑘𝑟^𝑘−1−1≥0. (2.9)
If (2.6) does not hold, then the numerator in (2.9) is negative for 𝑟 sufficiently close to 1. Therefore, there exists a point 𝑧₀=𝑟₀ in (0,1) for which the quotient in (2.9) is negative. This contradicts our assumption that 𝑓∈𝑉_𝐻(ℓ,𝑚,𝑘,𝜆₁,𝜆₂). We thus conclude that it is both necessary and sufficient that the coefficient bound inequality (2.6) holds true when 𝑓∈𝑉_𝐻(ℓ,𝑚,𝑘,𝜆₁,𝜆₂). This completes the proof of Theorem 2.2.

Theorem 2.3. The closed convex hull of 𝑓∈𝑉_𝐻(ℓ,𝑚,𝑘,𝜆₁,𝜆₂) (denoted by clco 𝑉_𝐻(ℓ,𝑚,𝑘,𝜆₁,𝜆₂)) is 𝑓(𝑧)=𝑧+^∞_𝑘=2𝑎_𝑘𝑧^𝑘+^∞_𝑘=1𝑏_𝑘𝑧^𝑘∶^∞_𝑘=2𝑘𝑎_𝑘+𝑏_𝑘<1−𝑏₁. (2.10) By setting 𝜆_𝑘=(1−ℓ)/((2𝑘−1−ℓ)((1+(𝜆₁+𝜆₂)(𝑘−1))/(1+𝜆₂(𝑘−1)))^𝑚) and 𝜇_𝑘=(1+ℓ)/((2𝑘+1+ℓ)  ((1+(𝜆₁+𝜆₂)(𝑘−1))/(1+𝜆₂(𝑘−1)))^𝑚), then for 𝑏₁ fixed, the extreme points for clco 𝑉_𝐻(ℓ,𝑚,𝑘,𝜆₁,𝜆₂) are 𝑧+𝜆_𝑘𝑥𝑧^𝑘+𝑏₁𝑧∪𝑧+𝑏₁𝑧+𝜇_𝑘𝑥𝑧^𝑘, (2.11) where 𝑘≥2 and |𝑥|=1−|𝑏₁|.

Proof. Any function 𝑓 in clco 𝑉_𝐻(ℓ,𝑚,𝑘,𝜆₁,𝜆₂) may be expressed as 𝑓(𝑧)=𝑧+^∞_𝑘=2𝑎_𝑘𝑒^𝑖𝜂𝑘𝑧^𝑘+𝑏₁𝑧+^∞_𝑘=2𝑏_𝑘𝑒^𝑖𝛿𝑘𝑧^𝑘, (2.12) where the coefficients satisfy the inequality (2.1). Set ℎ₁(𝑧)=𝑧, 𝑔₁(𝑧)=𝑏₁𝑧, ℎ_𝑘(𝑧)=𝑧+𝜆_𝑘𝑒^𝑖𝜂𝑘𝑧^𝑘, 𝑔_𝑘(𝑧)=𝑏₁𝑧+𝜇_𝑘𝑒^𝑖𝛿𝑘𝑧^𝑘 for 𝑘=2,3,…. Writing 𝜒_𝑘=|𝑎_𝑘|/𝜆_𝑘, 𝑌_𝑘=|𝑏_𝑘|/𝜇_𝑘, 𝑘=2,3,… and 𝜒₁=1−∑^∞_𝑘=2𝜒_𝑘; 𝑌₁=1−∑^∞_𝑘=2𝑌_𝑘, we get 𝑓(𝑧)=^∞_𝑘=1𝜒_𝑘ℎ_𝑘(𝑧)+𝑌_𝑘𝑔_𝑘(𝑧). (2.13) In particular, setting 𝑓₁(𝑧)=𝑧+𝑏₁𝑓𝑧,_𝑘(𝑧)=𝑧+𝜆_𝑘𝑥𝑧^𝑘+𝑏₁𝑧+𝜇𝑘𝑦𝑧^𝑘,𝑘≥2,|𝑥|+𝑦=1−𝑏₁. (2.14) We see that extreme points of clco 𝑓∈𝑉_𝐻(ℓ,𝑚,𝑘𝜆₁,𝜆₂)⊂{𝑓_𝑘(𝑧)}.
To see that 𝑓₁(𝑧) is not in extreme point, note that 𝑓₁(𝑧) may be written as 𝑓₁(𝑧)=12𝑓₁(𝑧)+𝜆₂1−𝑏₁𝑧²+12𝑓₁(𝑧)−𝜆₂1−𝑏₁𝑧², (2.15) a convex linear combination of functions in clco 𝑉_𝐻(ℓ,𝑚,𝑘,𝜆₁,𝜆₂).
To see that 𝑓_𝑚 is not an extreme point if both |𝑥|≠0 and |𝑦|≠0, we will show that it can then also be expressed as a convex linear combinations of functions in clco 𝑉_𝐻(ℓ,𝑚,𝑘,𝜆₁,𝜆₂). Without loss of generality, assume |𝑥|≥|𝑦|. Choose 𝜖>0 small enough so that 𝜖>|𝑥|/|𝑦|. Set 𝐴=1+𝜖 and 𝐵=1−|𝜖𝑥/𝑦|. We then see that both𝑡₁(𝑧)=𝑧+𝜆_𝑘𝐴𝑥𝑧^𝑘+𝑏₁𝑧+𝜇_𝑘𝑦𝐵𝑧^𝑘,𝑡₂(𝑧)=𝑧+𝜆_𝑘(2−𝐴)𝑥𝑧^𝑘+𝑏₁𝑧+𝜇_𝑘𝑦(2−𝐵)𝑧^𝑘 (2.16) are in clco 𝑉_𝐻(ℓ,𝑚,𝑘,𝜆₁,𝜆₂) and that 𝑓_𝑘(𝑧)=12𝑡₁(𝑧)+𝑡₂(𝑧). (2.17) The extremal coefficient bounds show that functions of the form (2.11) are the extreme points for clco 𝑉_𝐻(ℓ,𝑚,𝑘,𝜆₁,𝜆₂), and so the proof is complete.

Following Avici and Zlotkiewicz [5] and [6], we refer to the 𝛿-neighborhood of the functions 𝑓(𝑧) defined by (1.2) to be the set of functions 𝐹 for which 𝑁_𝛿𝑓=𝐹(𝑧)=𝑧+^∞_𝑘=2𝐴_𝑘𝑧^𝑘+^∞_𝑘=1𝐵_𝑘𝑧^𝑘,^∞_𝑘=2𝑘𝑎_𝑘−𝐴_𝑘+𝑏_𝑘−𝐵_𝑘+𝑏₁−𝐵₁≤𝛿. (2.18)

In our case, let us define the generalized 𝛿-neighborhood of 𝑓 to be the set 𝑁_𝛿𝑓=𝐹(𝑧)∶^∞_𝑘=21+𝜆₁+𝜆₂(𝑘−1)1+𝜆₂(𝑘−1)^𝑚(2𝑘−1−ℓ)𝑎_𝑘−𝐴_𝑘+(2𝑘+1+ℓ)𝑏_𝑘−𝐵_𝑘+(1−ℓ)𝑏₁−𝐵₁≤(1−ℓ)𝛿. (2.19)

Theorem 2.4. Let 𝑓 be given by (1.2). If 𝑓 satisfies the conditions ^∞_𝑘=2𝑘(2𝑘−1−ℓ)𝑎_𝑘1+𝜆₁+𝜆₂(𝑘−1)1+𝜆₂(𝑘−1)^𝑚+^∞_𝑘=1𝑘(2𝑘+1+ℓ)𝑏_𝑘1+𝜆₁+𝜆₂(𝑘−1)1+𝜆₂(𝑘−1)^𝑚≤(1−ℓ), (2.20) where 0≤ℓ<1, and 𝛿=1−ℓ3−ℓ1−3+ℓ𝑏1−ℓ¹, (2.21) then 𝑁(𝑓)⊂𝐺_𝐻(ℓ,𝑚,𝑘,𝜆₁,𝜆₂).

Proof. Let 𝑓 satisfy (2.20) and 𝐹(𝑧) be given by 𝐹(𝑧)=𝑧+𝐵₁𝑧+^∞_𝑘=2𝐴_𝑘𝑧^𝑚+𝐵_𝑘𝑧^𝑘 (2.22) which belong to 𝑁(𝑓). We obtain (3+ℓ)𝐵₁+^∞_𝑘=2(2𝑘−1−ℓ)𝐴_𝑘+(2𝑘+1+ℓ)𝐵_𝑘1+𝜆₁+𝜆₂(𝑘−1)1+𝜆₂(𝑘−1)^𝑚≤(3+ℓ)𝐵₁−𝑏₁+(3+ℓ)𝑏₁+^∞_𝑘=21+𝜆₁+𝜆₂(𝑘−1)1+𝜆₂(𝑘−1)^𝑚(2𝑘−1−ℓ)𝐴_𝑘−𝑎_𝑘+(2𝑘+1+ℓ)𝐵_𝑘−𝑏_𝑘+^∞_𝑘=21+𝜆₁+𝜆₂(𝑘−1)1+𝜆₂(𝑘−1)^𝑚(2𝑘−1−ℓ)𝑎_𝑘+(2𝑘+1+ℓ)𝑏_𝑘≤(1−ℓ)𝛿+(3+ℓ)𝑏₁+13−ℓ^∞_𝑘=2𝑘1+𝜆₁+𝜆₂(𝑘−1)1+𝜆₂(𝑘−1)^𝑚(2𝑘−1−ℓ)𝑎_𝑘+(2𝑘+1+ℓ)𝑏_𝑘≤(1−ℓ)𝛿+(3+ℓ)𝑏₁+1)−(𝑏3−ℓ(1−ℓ3+ℓ)₁≤1−ℓ. (2.23) Hence for 𝛿=(1−ℓ)/(3−ℓ)(1−((3+ℓ)/(1−ℓ))|𝑏₁|), we infer that 𝐹(𝑧)∈𝐺_𝐻(ℓ,𝑚,𝑘,𝜆₁,𝜆₂) which concludes the proof of Theorem 2.4.

Acknowledgment

The work here was supported by UKM-ST-06-FRGS0244-2010.