The Generalized Janowski Starlike and Close-to-Starlike Log-Harmonic Mappings

Abstract

Motivated by the success of the Janowski starlike function, we consider here closely related functions for log-harmonic mappings of the form 𝑓(𝑧)=𝑧ℎ(𝑧)𝑔(𝑧) defined on the open unit disc 𝑈. The functions are in the class of the generalized Janowski starlike log-harmonic mapping, 𝑆^∗_lh(𝐴,𝐵,𝛼), with the functional 𝑧ℎ(𝑧) in the class of the generalized Janowski starlike functions, 𝑆^∗(𝐴,𝐵,𝛼). By means of these functions, we obtained results on the generalized Janowski close-to-starlike log-harmonic mappings, 𝐶𝑆𝑇_lh(𝐴,𝐵,𝛼).

1. Introduction

The class 𝑆^∗(𝐴,𝐵) was investigated by Janowski [1] in early 1970, and since then various other subclasses in relation with this Janowski class have been introduced and studied. In that direction, the log-harmonic mappings which have been studied extensively for the past 3 decades, (see [2–10]) were also associated with the Janowski class. Perhaps, the Janowski starlike log-harmonic univalent functions were first introduced by Polatoğlu and Deniz [11].

A function 𝑓 is said to be log-harmonic on the open unit disc 𝑈={𝑧∶|𝑧|<1} if it satisfies the nonlinear elliptic partial differential equation: 𝑓_𝑧𝑓𝑓=𝑎^𝑧𝑓, (1.1) where the second dilatation function 𝑎∈ℋ(𝑈) (set of all analytic functions defined on 𝑈) such that |𝑎(𝑧)|<1 for all 𝑧∈𝑈. For analytic functions ℎ and 𝑔 in 𝑈, the function 𝑓 can be expressed as𝑓(𝑧)=ℎ(𝑧)𝑔(𝑧) (1.2) if 𝑓 is a nonvanishing log-harmonic mapping and𝑓(𝑧)=𝑧|𝑧|^2𝛽ℎ(𝑧)𝑔(𝑧) (1.3) if 𝑓 vanishes at 𝑧=0 but is not identically zero (for Re𝛽>−1/2, 𝑔(0)=1, and ℎ(0)≠0).

Let 𝑓(𝑧)=𝑧ℎ(𝑧)𝑔(𝑧) be a univalent log-harmonic mapping, where 0∉𝑓(𝑈) or equivalently 0∉ℎ𝑔(𝑈). Then 𝑓 is starlike log-harmonic mapping ifRe𝑧𝑓_𝑧−𝑧𝑓_𝑧𝑓>0. (1.4) Results on starlike log-harmonic mapping of order 𝛼 was given in [6].

Motivated by [11], the class of the generalized Janowski log-harmonic starlike functions was introduced in [12]. For real numbers 𝐴 and 𝐵, with −1≤𝐵<𝐴≤1 and 0≤𝛼<1, the family of analytic functions of the form𝑝(𝑧)=1+𝑝₁𝑧+𝑝₂𝑧²+⋯ (1.5) is in 𝑃(𝐴,𝐵,𝛼) if and only if𝑝(𝑧)=1+[(1−𝛼)𝐴+𝛼𝐵]𝜙(𝑧)1+𝐵𝜙(𝑧), (1.6) where the function 𝜙 is analytic in 𝑈 with 𝜙(0)=0 and |𝜙(𝑧)|<1. The following lemma is also essential for 𝑝(𝑧) to be in 𝑃(𝐴,𝐵,𝛼).

Lemma 1.1 (see [13]). The function 𝑝(𝑧)∈𝑃(𝐴,𝐵,𝛼) if and only if 𝑝(𝑧)−1−[(1−𝛼)𝐴+𝛼𝐵]𝐵𝑟²1−𝐵²𝑟²≤(1−𝛼)(𝐴−𝐵)𝑟1−𝐵²𝑟² (1.7) for |𝑧|≤𝑟<1.

Let 𝑆^∗(𝐴,𝐵,𝛼) denote the class of the generalized Janowski starlike functions of the analytic functions 𝑠(𝑧)=𝑧+𝑠₂𝑧²+⋯ such that 𝑠(𝑧)∈𝑆^∗(𝐴,𝐵,𝛼) if and only if𝑧𝑠^′(𝑧)𝑠(𝑧)=𝑝(𝑧) (1.8) and 𝑝(𝑧)∈𝑃(𝐴,𝐵,𝛼) for 𝑧∈𝑈.

For univalent log-harmonic mapping 𝑓(𝑧)=𝑧ℎ(𝑧)𝑔(𝑧) with 𝑔(0)=1 and ℎ(0)≠0, 𝑓 is in the class of the generalized Janowski starlike log-harmonic mapping denoted by 𝑆^∗_lh(𝐴,𝐵,𝛼) if𝑝(𝑧)−1−[(1−𝛼)𝐴+𝛼𝐵]𝐵𝑟²1−𝐵²𝑟²≤(1−𝛼)(𝐴−𝐵)𝑟1−𝐵²𝑟², (1.9) where𝑝(𝑧)=ℎ(𝑧)𝑔(𝑧)+𝑧ℎ^′(𝑧)𝑔(𝑧)−𝑧𝑔^′(𝑧)ℎ(𝑧)ℎ(𝑧)𝑔(𝑧)=1+𝑧ℎ^′(𝑧)ℎ(𝑧)−𝑧𝑔^′(𝑧)𝑔(𝑧). (1.10) Also observe that if 𝑓∈𝑆^∗_lh(𝐴,𝐵,𝛼), thenRe𝑧𝑓_𝑧−𝑧𝑓_𝑧𝑓≥1−[(1−𝛼)𝐴+𝛼𝐵]1−𝐵. (1.11)

In the present work, we consider the log-harmonic mapping 𝑓(𝑧)=𝑧ℎ(𝑧)𝑔(𝑧) in the generalized Janowski starlike functions with the functional 𝑧ℎ(𝑧)∈𝑆^∗(𝐴,𝐵,𝛼). We also study the class of generalized Janowski close-to-starlike in the next section.

2. The Generalized Janowski Starlike Log-Harmonic

Theorem 2.1. If 𝑧ℎ(𝑧)∈𝑆^∗(𝐴,𝐵,𝛼), then (1−𝐵𝑟)^{(1−𝛼)(𝐴−𝐵)/𝐵}≤|ℎ(𝑧)|≤(1+𝐵𝑟)^{(1−𝛼)(𝐴−𝐵)/𝐵}𝑒for𝐵≠0,_{−(1−𝛼)𝐴𝑟}≤|ℎ(𝑧)|≤𝑒^{(1−𝛼)𝐴𝑟}for𝐵=0. (2.1)

Proof. Since 𝑧ℎ(𝑧)∈𝑆^∗(𝐴,𝐵,𝛼), Lemma 1.1 yields that for 𝐵≠0 we have 1−[(1−𝛼)𝐴+𝛼𝐵]𝑟𝑧((𝑧1−𝐵𝑟≤Re𝑧ℎ))^′(𝑧)≤)]𝑟𝑧ℎ1+[(1−𝛼𝐴+𝛼𝐵1+𝐵𝑟 (2.2) or −(1−𝛼)(𝐴−𝐵)𝑟1−𝐵𝑟≤Re𝑧ℎ^′(𝑧)ℎ(𝑧)≤(1−𝛼)(𝐴−𝐵)𝑟1+𝐵𝑟. (2.3) Simple calculations yield −(1−𝛼)(𝐴−𝐵)()|ℎ(𝑧≤()−𝐵log1−𝐵𝑟≤log)|1−𝛼)(𝐴−𝐵𝐵log(1+𝐵𝑟), (2.4) and the result follows immediately.
For 𝐵=0, Lemma 1.1 yields ()𝑧((𝑧1−1−𝛼𝐴𝑟≤Re𝑧ℎ))^′(𝑧)()𝑧ℎ≤1+1−𝛼𝐴𝑟, (2.5) and the proof is completed similarly.

Theorem 2.2. Let 𝑓(𝑧)=𝑧ℎ(𝑧)𝑔(𝑧)∈𝑆^∗_lh(𝐴,𝐵,𝛼) with 𝑧ℎ(𝑧)∈𝑆^∗(𝐴,𝐵,𝛼). Then one has (1−𝐵𝑟)^{(1−𝛼)(𝐴−𝐵)/𝐵}(1+𝐵𝑟)^{(1−𝛼)(𝐴−𝐵)/𝐵}≤𝑔(𝑧≤())1+𝐵𝑟^{(1−𝛼)(𝐴−𝐵)/𝐵}(1−𝐵𝑟)^{(1−𝛼)(𝐴−𝐵)/𝐵}𝑒for𝐵≠0,_{−2(1−𝛼)𝐴𝑟}≤𝑔(𝑧)≤𝑒^{2(1−𝛼)𝐴𝑟}for𝐵=0. (2.6)

Proof. It follows from [12] that for 𝑓(𝑧)=𝑧ℎ(𝑧)𝑔(𝑧)∈𝑆^∗_lh(𝐴,𝐵,𝛼), we have (1−𝐵𝑟)^{(1−𝛼)(𝐴−𝐵)/𝐵}≤ℎ(𝑧)𝑔(𝑧)≤(1+𝐵𝑟)^{(1−𝛼)(𝐴−𝐵)/𝐵}𝑒for𝐵≠0,_{−(1−𝛼)𝐴𝑟}≤ℎ(𝑧)𝑔(𝑧)≤𝑒^{(1−𝛼)𝐴𝑟}for𝐵=0. (2.7) With these inequalities and Theorem 2.1, we can conclude the following statement.

Theorem 2.3. Let 𝑓(𝑧)=𝑧ℎ(𝑧)𝑔(𝑧)∈𝑆^∗_lh(𝐴,𝐵,𝛼) with 𝑧ℎ(𝑧)∈𝑆^∗(𝐴,𝐵,𝛼). Then one has 𝑟(1−𝐵𝑟)^{2(1−𝛼)(𝐴−𝐵)/𝐵}(1+𝐵𝑟)^{(1−𝛼)(𝐴−𝐵)/𝐵}≤𝑓(𝑧≤𝑟())1+𝐵𝑟^{2(1−𝛼)(𝐴−𝐵)/𝐵}(1−𝐵𝑟)^{(1−𝛼)(𝐴−𝐵)/𝐵}for𝐵≠0,𝑟𝑒^{−3(1−𝛼)𝐴𝑟}≤𝑓(𝑧)≤𝑟𝑒^{3(1−𝛼)𝐴𝑟}for𝐵=0. (2.8)

Proof. For 𝑓(𝑧)=𝑧ℎ(𝑧)𝑔(𝑧) and |𝑧|=𝑟, it is easy to see that 𝑓(𝑧)=𝑧ℎ(𝑧)𝑔(𝑧)=|𝑧||ℎ(𝑧)|𝑔(𝑧)=𝑟|ℎ(𝑧)|𝑔(𝑧). (2.9) Thus, we can obtain the results from Theorems 2.1 and 2.2.

3. The Generalized Janowski Close-to-Starlike Log-Harmonic

Let 𝑃_lh be mapping the set of all log-harmonic mappings, and let 𝑅 be defined on 𝑈 which are of the form 𝑅(𝑧)=𝐾(𝑧)𝐽(𝑧), where 𝐾 and 𝐽 are in ℋ(𝑈), 𝐾(0)=𝐽(0)=1 and such that Re𝑅(𝑧)>0 for all 𝑧∈𝑈. These log-harmonic mappings with positive real part were studied in [5]. Other interesting studies in the same paper were on the close-to starlike log-harmonic mappings. The author then extended the results to close-to starlike of order 𝛼 log-harmonic mappings [2].

In that direction, we say that 𝐹(𝑧)=𝑧𝐻(𝑧)𝐺(𝑧) is the generalized Janowski close-to-starlike log-harmonic mapping if there exist a log-harmonic mapping 𝑓(𝑧)=𝑧ℎ(𝑧)𝑔(𝑧)∈𝑆𝑇^∗_lh(𝐴,𝐵,𝛼) (−1≤𝐵<𝐴≤1 and 0≤𝛼<1), with respect to the second dilatation function 𝑎∈ℋ(𝑈) and a log-harmonic mapping with positive real part 𝑅∈𝑃_lh where its second dilatation function is the same such that𝐹(𝑧)=𝑓(𝑧)𝑅(𝑧) (3.1) or equivalently𝐹(𝑧)Re𝑓(𝑧)>0. (3.2) We could also easily derive from (3.1) thatRe𝑧𝐹_𝑧−𝑧𝐹_𝑧𝐹=Re𝑧𝑓_𝑧−𝑧𝑓_𝑧𝑓+Re𝑧𝑅_𝑧−𝑧𝑅_𝑧𝑅. (3.3)

The geometrical interpretation is that under a generalized Janowski close-to-starlike log-harmonic mapping, the radius vector of the image of |𝑧|=𝑟<1 never turns back by the amount more than ((1−𝛼)(𝐴−𝐵)/(1−𝐵))𝜋. As special cases, we see that(i)for 𝛼=0 or under the Janowski close-to-starlike log-harmonic mappings, the radius vector of the image of |𝑧|=𝑟<1 never turns back by an amount more than ((𝐴−𝐵)/(1−𝐵))𝜋, (ii)for when 𝐴=1,  𝐵=−1 or under the close-to-starlike of order 𝛼 log-harmonic mappings, the radius vector of the image of |𝑧|=𝑟<1 never turns back by an amount more than (1−𝛼)𝜋, (iii)for 𝛼=0,  𝐴=1,  𝐵=−1 or under the close-to-starlike log-harmonic mappings, the radius vector of the image of |𝑧|=𝑟<1 never turns back by an amount more than 𝜋.

The following theorem gives us the radius of starlikeness for 𝐹(𝑧)=𝑧𝐻(𝑧)𝐺(𝑧)∈𝐶𝑆𝑇_lh(𝐴,𝐵,𝛼).

Theorem 3.1. The radius of starlikeness for 𝐹(𝑧)=𝑧𝐻(𝑧)𝐺(𝑧)∈𝐶𝑆𝑇_lh(𝐴,𝐵,𝛼) is the largest positive root, 𝑟∈(0,1], such that (1−[(1−𝛼)𝐴+𝛼𝐵]𝑟)(1−𝑟)(1+𝑟)−2𝑟(1−𝐵𝑟)>0. (3.4)

Proof. For 𝐹(𝑧)=𝑧𝐻(𝑧)𝐺(𝑧)∈𝐶𝑆𝑇_lh(𝐴,𝐵,𝛼), we have Re𝑧𝐹_𝑧−𝑧𝐹_𝑧𝐹=Re𝑧𝑓_𝑧−𝑧𝑓_𝑧𝑓+Re𝑧𝑅_𝑧−𝑧𝑅_𝑧𝑅, (3.5) and since 𝑓∈𝑆^∗_lh(𝐴,𝐵,𝛼) and 𝑅∈𝑃_lh, (3.5) becomes Re𝑧𝐹_𝑧−𝑧𝐹_𝑧𝐹≥1−[(1−𝛼)𝐴+𝛼𝐵]𝑟+1−𝐵𝑟−2𝑟1−𝑟². (3.6) Hence, Re𝑧𝐹_𝑧−𝑧𝐹_𝑧𝐹>0 (3.7) if 1−[(1−𝛼)𝐴+𝛼𝐵]𝑟−1−𝐵𝑟2𝑟1−𝑟²>0. (3.8)

Corollary 3.2 (see [2]). The radius of starlikeness for 𝐹(𝑧)=𝑧𝐻(𝑧)𝐺(𝑧)∈𝐶𝑆𝑇_lh is 𝑟<2−√3. (3.9)

Corollary 3.3 (see [2]). The radius of starlikeness for 𝐹(𝑧)=𝑧𝐻(𝑧)𝐺(𝑧)∈𝐶𝑆𝑇_lh(𝛼) is √𝑟<2−𝛼−𝛼²−2𝛼+31−2𝛼. (3.10)

Corollary 3.4. The radius of starlikeness for 𝐹(𝑧)=𝑧𝐻(𝑧)𝐺(𝑧)∈𝐶𝑆𝑇_lh(𝐴,𝐵) is the largest positive root, 𝑟∈(0,1], such that (1−𝐴𝑟)(1−𝑟)(1+𝑟)−2𝑟(1−𝐵𝑟)>0. (3.11)

Proof. The proof is completed by taking 𝛼=0 in (3.4).

We need the following theorem from [5] to prove our next result.

Theorem
Let 𝑅(𝑧)∈𝑃_lh, and suppose that 𝑎(0)=0. Then, for 𝑧∈𝑈, we have 𝑒^{−2|𝑧|/(1−|𝑧|)}≤|𝑅(𝑧)|≤𝑒^{2|𝑧|/(1−|𝑧|)}. (3.12)

Theorem 3.5. For 𝐹(𝑧)=𝑧𝐻(𝑧)𝐺(𝑧)∈𝐶𝑆𝑇_lh(𝐴,𝐵,𝛼) and 𝑓(𝑧)=𝑧ℎ(𝑧)𝑔(𝑧) with 𝑧ℎ(𝑧)∈𝑆^∗(𝐴,𝐵,𝛼), one has 𝑟(1−𝐵𝑟)^{2(1−𝛼)(𝐴−𝐵)/𝐵}𝑒^{−2𝑟/(1−𝑟)}(1+𝐵𝑟)^{(1−𝛼)(𝐴−𝐵)/𝐵}≤|𝐹(𝑧≤𝑟())|1−𝐵𝑟^{2(1−𝛼)(𝐴−𝐵)/𝐵}𝑒^{2𝑟/(1−𝑟)}(1+𝐵𝑟)^{(1−𝛼)(𝐴−𝐵)/𝐵}for𝐵≠0,𝑟𝑒^{−3(1−𝛼)𝐴𝑟−(2𝑟/(1−𝑟))}≤|𝐹(𝑧)|≤𝑟𝑒^{3(1−𝛼)𝐴𝑟−(2𝑟/(1−𝑟))}for𝐵=0. (3.13)

Proof. From (3.12) and Theorem 2.3, we have 𝑒^{−2𝑟/(1−𝑟)}≤|𝑅(𝑧)|𝑒^{2𝑟/(1−𝑟)},|𝑧|𝑟()=𝑟<1,1−𝐵𝑟^{2(1−𝛼)(𝐴−𝐵)/𝐵}(1+𝐵𝑟)^{(1−𝛼)(𝐴−𝐵)/𝐵}≤𝑓(𝑧≤𝑟())1+𝐵𝑟^{2(1−𝛼)(𝐴−𝐵)/𝐵}(1−𝐵𝑟)^{(1−𝛼)(𝐴−𝐵)/𝐵}for𝐵≠0,𝑟𝑒^{−3(1−𝛼)𝐴𝑟}≤𝑓(𝑧)≤𝑟𝑒^{3(1−𝛼)𝐴𝑟}for𝐵=0, (3.14) respectively. Also, we know that for 𝐹∈𝐶𝑆𝑇_lh(𝐴,𝐵,𝛼), we have 𝐹(𝑧)=𝑓(𝑧)𝑅(𝑧) which then leads to the desired result.

Acknowledgment

The work presented here was partially supported by UKM-ST-FRGS-0244-2010.