Certain New Classes of Analytic Functions with Varying Arguments

Abstract

We introduce certain new classes 𝜅−𝑉ST(𝛼,𝛽) and 𝜅−𝑉UCV(𝛼,𝛽), which represent the κ uniformly starlike functions of order α and type β with varying arguments and the κ uniformly convex functions of order α and type β with varying arguments, respectively. Moreover, we give coefficients estimates, distortion theorems, and extreme points of these classes.

1. Introduction

Let 𝒜 denote the class of functions of the following form: 𝑓(𝑧)=𝑧+^∞_𝑛=2𝑎_𝑛𝑧^𝑛, (1) that are analytic and univalent in the open unit disc 𝑈={𝑧∈ℂ∶|𝑧|<1}.

Definition 1 (see [1]). Let 𝜅−ST(𝛼,𝛽) denote the subclass of 𝒜 consisting of functions 𝑓(𝑧) of the form (1) and satisfy the following inequality: Re𝑧𝑓^′(𝑧)𝑓(𝑧)−𝛼>𝜅𝑧𝑓^′(𝑧)𝑓(𝑧),<().−𝛽0≤𝛼<𝛽≤1;𝜅1−𝛽1−𝛼;𝑧∈𝑈 (2) Also let 𝜅−UCV(𝛼,𝛽) denote the subclass of 𝒜 consisting of functions 𝑓(𝑧) of the form (1) and satisfy the following inequality: Re1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)−𝛼>𝜅1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧),<().−𝛽0≤𝛼<𝛽≤1;𝜅1−𝛽1−𝛼;𝑧∈𝑈 (3) It follows from (2) and (3) that 𝑓∈𝜅−UCV𝛼,𝛽⟺𝑧𝑓^′∈𝜅−ST𝛼,𝛽. (4) The class 𝜅−ST(𝛼,𝛽) denote the class of 𝜅 uniformly starlike functions of order 𝛼 and type 𝛽 and the class 𝜅−UCV(𝛼,𝛽) denotes the class of 𝜅 uniformly convex functions of order 𝛼 and type 𝛽.

Specializing parameters 𝛽,  𝛼, and 𝜅, we obtain the following subclasses studied by various authors:(i)𝜅−ST(𝛼,1)=SD(𝜅,𝛼) and 𝜅−UCV(𝛼,1)=KD(𝜅,𝛼) (see [2, 3]);(ii)1−ST(𝛼,1)=𝑆_𝑝(𝛼) and 1−UCV(𝛼,1)=UCV(𝛼) (see [4]);(iii)𝜅−ST(0,1)=𝜅−ST and 𝜅−UCV(0,1)=𝜅−UCV (see [5, 6]);(iv)1−ST(0,1)=𝑆_𝑝 and 1−UCV(0,1)=UCV (see [4, 7–10]).

Also we note that =𝑓(𝑧)1−ST𝛼,𝛽=US𝛼,𝛽∈𝒜∶Re𝑧𝑓^′(𝑧)𝑓(𝑧)−𝛼>𝑧𝑓^′(𝑧)𝑓(𝑧),=𝑓(𝑧)−𝛽1−UCV𝛼,𝛽=UC𝛼,𝛽∈𝒜∶Re1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)−𝛼>1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)−𝛽, (5) which are the uniformly starlike functions of order 𝛼 and type 𝛽 and uniformly convex functions of order 𝛼 and type 𝛽, respectively.

Definition 2 (see [11]). A function 𝑓(𝑧) of the form (1) is said to be in the class 𝑉(𝜃_𝑛) if 𝑓∈𝒜 and arg(𝑎_𝑛)=𝜃_𝑛 for all 𝑛≥2. If furthermore there exist a real number 𝛿 such that 𝜃_𝑛+(𝑛−1)𝛿≡𝜋(mod2𝜋), then 𝑓(𝑧) is said to be in the class 𝑉(𝜃_𝑛,𝛿). The union of 𝑉(𝜃_𝑛,𝛿) taken over all possible sequences {𝜃_𝑛} and all possible real numbers 𝛿 is denoted by 𝑉.
Let 𝜅−𝑉ST(𝛼,𝛽) denote the subclass of 𝑉 consisting of functions 𝑓(𝑧)∈𝜅−ST(𝛼,𝛽). Also Let 𝜅−𝑉UCV(𝛼,𝛽) denote the subclass of 𝑉 consisting of functions 𝑓(𝑧)∈𝜅−UCV(𝛼,𝛽).

In this paper we obtain coefficient bounds for functions in the classes 𝜅−𝑉ST(𝛼,𝛽) and 𝜅−𝑉UCV(𝛼,𝛽), respectively, further we obtain distortion bounds and the extreme points for functions in these classes.

2. Coefficient Estimates

Unless otherwise mentioned, we assume in the reminder of this paper that 0≤𝛼<𝛽≤1, 𝜅(1−𝛽)<(1−𝛼) and 𝑧∈𝑈.

We shall need the following lemmas.

Lemma 3. The sufficient condition for 𝑓(𝑧)∈𝒜 given by (1) to be in the class 𝜅−ST(𝛼,𝛽) is that ^∞_𝑛=2𝜅𝑛−𝛽+(𝑛−𝛼)𝑎_𝑛≤1−𝛼−𝜅1−𝛽. (6)

Proof. It suffices to show that inequality (2) holds true. Upon using the fact that Re{𝑤}>𝜅𝑤−𝛽+𝛼⟺Re1+𝜅𝑒^𝑖𝜃𝑤−𝛽𝜅𝑒^𝑖𝜃>𝛼, (7) then inequality (2) may be written as Re1+𝜅𝑒^𝑖𝜃𝑧𝑓^′(𝑧)𝑓(𝑧)−𝛽𝜅𝑒^𝑖𝜃>𝛼, (8) or 𝐴(𝑧)Re𝐵(𝑧)>𝛼, (9) where 𝐴(𝑧)=(1+𝜅𝑒^𝑖𝜃)𝑧𝑓^′(𝑧)−𝛽𝜅𝑒^𝑖𝜃𝑓(𝑧) and 𝐵(𝑧)=𝑓(𝑧), then condition (2) or (9) is equivalent to |𝐴(𝑧)+(1−𝛼)𝐵(𝑧)|−|𝐴(𝑧)−(1+𝛼)𝐵(𝑧)|≥0. (10) We note that |𝐴(z)+()𝐵(𝑧=1−𝛼)|1−𝛽𝜅𝑒^𝑖𝜃𝑧−+2−𝛼_∞𝑛=2𝛽−𝑛𝜅𝑒^𝑖𝜃+𝛼−𝑛−1𝑎_𝑛𝑧^𝑛≥𝑧|−−𝜅1−𝛽+2−𝛼|_∞𝑛=2𝑛−𝛽𝜅+𝑛−𝛼+1𝑎_𝑛|𝑧|^𝑛,|𝐴(𝑧)−()𝐵(𝑧=1+𝛼)|1−𝛽𝜅𝑒^𝑖𝜃𝑧+−𝛼_∞𝑛=2𝑛−𝛽𝜅𝑒^𝑖𝜃+𝑛−𝛼−1𝑎_𝑛𝑧^𝑛≤𝜅𝑧|+1−𝛽+𝛼|_∞𝑛=2𝑛−𝛽𝜅+𝑛−𝛼−1𝑎_𝑛|𝑧|^𝑛. (11)(12) Using (11) and (12), then we can obtain the following inequality: |𝐴(𝑧)+()𝐵(𝑧−|𝐴(𝑧)−()𝐵(𝑧)𝑧|1−𝛼)|1+𝛼)|≥2(1−𝛼−𝜅1−𝛽|−2^∞_𝑛=2𝑛−𝛽𝜅+𝑛−𝛼𝑎_𝑛|𝑧|^𝑛. (13) The expression |𝐴(𝑧)+(1−𝛼)𝐵(𝑧)|−|𝐴(𝑧)−(1+𝛼)𝐵(𝑧)| is bounded below by 0 if 2(1−𝛼)−1−𝛽𝜅|𝑧|−2^∞_𝑛=2𝑛−𝛽𝜅+𝑛−𝛼𝑎_𝑛|𝑧|^𝑛>0, (14) or ^∞_𝑛=2𝜅𝑛−𝛽+(𝑛−𝛼)𝑎_𝑛<1−𝛼−𝜅1−𝛽. (15) Hence the proof of Lemma 3 is completed.

By using (4) and (6) we can obtain the following lemma.

Lemma 4. A function 𝑓(𝑧) of the form (1) is in the class 𝜅−UCV(𝛼,𝛽) if ^∞_𝑛=2𝑛𝜅𝑛−𝛽+(𝑛−𝛼)𝑎_𝑛<1−𝛼−𝜅1−𝛽. (16)

Remark 5. Putting 𝛽=1 in Lemmas 3 and 4, we obtain the results obtained by Shams et al. [3, Theorems 2.1, 2.2, resp.].

In the following theorems, we show that the conditions (6) and (16) are also necessary for functions 𝑓(𝑧)∈𝜅−𝑉ST(𝛼,𝛽) and 𝜅−𝑉UCV(𝛼,𝛽), respectively.

Theorem 6. Let 𝑓(𝑧) be of the form (1), then 𝑓(𝑧)∈𝜅−𝑉ST(𝛼,𝛽) if and only if ^∞_𝑛=2𝜅𝑛−𝛽+(𝑛−𝛼)𝑎_𝑛<1−𝛼−𝜅1−𝛽. (17)

Proof. In view of Lemma 3, we need only to show that function 𝑓(𝑧)∈𝜅−𝑉ST(𝛼,𝛽) satisfies the coefficient inequality (17). If 𝑓(𝑧)∈𝜅−𝑉ST(𝛼,𝛽), then from (2), we have Re𝑧𝑓^′(𝑧)𝑓(𝑧)−𝛼>𝜅𝑧𝑓^′(𝑧)𝑓(𝑧)−𝛽, (18) thus we have ()Re1−𝛼+∑^∞_𝑛=2(𝑛−𝛼)𝑎_𝑛𝑧^𝑛−11+∑^∞_𝑛=2𝑎_𝑛𝑧^𝑛−1>𝜅1−𝛽+∑^∞_𝑛=2𝑛−𝛽𝑎_𝑛𝑧^𝑛−11+∑^∞_𝑛=2𝑎_𝑛𝑧^𝑛−1. (19) Since 𝑓(𝑧)∈𝑉, 𝑓(𝑧) lies in the class 𝑉(𝜃_𝑛,𝛿) for some sequence {𝜃_𝑛} and a real number 𝛿 such that 𝜃_𝑛+(𝑛−1)𝛿≡𝜋(mod2𝜋). (20) Set 𝑧=𝑟𝑒^𝑖𝛿 in (19), then we obtain (1−𝛼)−∑^∞_𝑛=2(𝑛−𝛼)𝑎_𝑛𝑟^𝑛−11−∑^∞_𝑛=2𝑎_𝑛𝑟^𝑛−1+∑>𝜅1−𝛽^∞_𝑛=2𝑛−𝛽𝑎_𝑛𝑟^𝑛−11−∑^∞_𝑛=2𝑎_𝑛𝑟^𝑛−1. (21) Letting 𝑟→1, then we have ^∞_𝑛=2𝜅𝑛−𝛽+(𝑛−𝛼)𝑎_𝑛<1−𝛼−𝜅1−𝛽. (22) Hence the proof of Theorem 6 is completed.

Corollary 7. If 𝑓(𝑧)∈𝜅−𝑉ST(𝛼,𝛽), then 𝑎_𝑛≤1−𝛼−𝜅1−𝛽𝜅+()()𝑛−𝛽𝑛−𝛼𝑛≥2. (23) The equality holds for the function 𝑓(𝑧)=𝑧+1−𝛼−𝜅1−𝛽𝜅+()𝑒𝑛−𝛽𝑛−𝛼^𝑖𝜃𝑛𝑧^𝑛(𝑛≥2;𝑧∈𝑈). (24)

Using the same technique used in Theorem 6 we get the following theorem

Theorem 8. Let 𝑓(𝑧) of the form (1), then 𝑓(𝑧)∈𝜅−𝑉UCV(𝛼,𝛽) if and only if ^∞_𝑛=2𝑛𝜅𝑛−𝛽+(𝑛−𝛼)𝑎_𝑛<1−𝛼−𝜅1−𝛽. (25)

Corollary 9. If 𝑓(𝑧)∈𝜅−𝑉UCV(𝛼,𝛽), then 𝑎_𝑛≤1−𝛼−𝜅1−𝛽𝑛𝜅+(()𝑛−𝛽𝑛−𝛼)𝑛≥2. (26) The equality holds for the function 𝑓(𝑧)=𝑧+1−𝛼−𝜅1−𝛽𝑛𝜅+(𝑒𝑛−𝛽𝑛−𝛼)^𝑖𝜃𝑛𝑧^𝑛(𝑛≥2;𝑧∈𝑈). (27)

3. Distortion Theorems

Theorem 10. Let the function 𝑓(𝑧) defined by (1) be in the class 𝜅−𝑉ST(𝛼,𝛽). Then |𝑧|−1−𝛼−𝜅1−𝛽𝜅+()|𝑧|2−𝛽2−𝛼²≤𝑓(𝑧≤|𝑧|+)1−𝛼−𝜅1−𝛽𝜅+()|𝑧|2−𝛽2−𝛼². (28) The result is sharp.

Proof . We employ the same technique as used by Silverman [11]. In view of Theorem 6, since Φ(𝑛)=𝜅𝑛−𝛽+(𝑛−𝛼) (29) is an increasing function of 𝑛(𝑛≥2), we have Φ(2)^∞_𝑛=2𝑎_𝑛≤^∞_𝑛=2Φ(𝑛)𝑎_𝑛≤1−𝛼−𝜅1−𝛽, (30) that is, ^∞_𝑛=2𝑎_𝑛≤1−𝛼−𝜅1−𝛽Φ(2). (31) Thus we have 𝑓(𝑧)≤|𝑧|+|𝑧|^2∞_𝑛=2𝑎_𝑛,𝑓(𝑧)≤|𝑧|+1−𝛼−𝜅1−𝛽𝜅+()|𝑧|2−𝛽2−𝛼². (32) Similarly, we get 𝑓(𝑧)≥|𝑧|−^∞_𝑛=2𝑎_𝑛|𝑧|^𝑛≥|𝑧|−|𝑧|^2∞_𝑛=2𝑎_𝑛≥|𝑧|−1−𝛼−𝜅1−𝛽𝜅+()|𝑧|2−𝛽2−𝛼². (33) This completes the proof of Theorem 10. Finally the result is sharp for the following function: 𝑓(𝑧)=𝑧+1−𝛼−𝜅1−𝛽𝜅+()𝑒2−𝛽2−𝛼^𝑖𝜃2𝑧², (34) at 𝑧=±|𝑧|𝑒^−𝑖𝜃2.

Corollary 11. Under the hypotheses of Theorem 8, 𝑓(𝑧) is included in a disc with center at the origin and radius 𝑟₁ given by 𝑟₁=1+1−𝛼−𝜅1−𝛽𝜅2−𝛽+(2−𝛼). (35)

Theorem 12. Let the function 𝑓(𝑧) defined by (1) be in the class 𝜅−𝑉ST(𝛼,𝛽). Then 21−1−𝛼−𝜅1−𝛽𝜅+()|𝑧|≤𝑓2−𝛽2−𝛼_′(𝑧2)≤1+1−𝛼−𝜅1−𝛽𝜅+()|𝑧|.2−𝛽2−𝛼 (36) The result is sharp.

Proof. Similarly Φ(𝑛)/𝑛 is an increasing function of 𝑛(𝑛≥2), in view of Theorem 6, we have Φ(2)2^∞_𝑛=2𝑛𝑎_𝑛≤^∞_𝑛=2Φ(𝑛)𝑎_𝑛≤1−𝛼−𝜅1−𝛽, (37) that is, ^∞_𝑛=2𝑛𝑎_𝑛≤21−𝛼−𝜅1−𝛽Φ(2). (38) Thus we have 𝑓^′(𝑧)≤1+|𝑧|^∞_𝑛=2𝑛𝑎_𝑛2≤1+1−𝛼−𝜅1−𝛽𝜅+()|𝑧|.2−𝛽2−𝛼 (39) Similarly 𝑓^′(𝑧)≥1−|𝑧|^∞_𝑛=2𝑛𝑎_𝑛2≥1−1−𝛼−𝜅1−𝛽𝜅+()|𝑧|.2−𝛽2−𝛼 (40) Finally, we can see that the assertions of Theorem 12 are sharp for the function 𝑓(𝑧) defined by (34). This completes the proof of Theorem 12.

Corollary 13. Under the hypotheses of Theorem 12, 𝑓^′(𝑧) is included in a disc with center at the origin and radius 𝑟₂ given by 𝑟₂2=1+1−𝛼−𝜅1−𝛽𝜅2−𝛽+(2−𝛼). (41)

Using the same technique used in Theorems 10 and 12, we get the following theorems.

Theorem 14. Let the function 𝑓(𝑧) defined by (1) be in the class 𝜅−𝑉UCV(𝛼,𝛽). Then |𝑧|−1−𝛼−𝜅1−𝛽2𝜅+(|𝑧|2−𝛽2−𝛼)²≤𝑓(𝑧≤|𝑧|+)1−𝛼−𝜅1−𝛽2𝜅+(|𝑧|2−𝛽2−𝛼)². (42) The result is sharp for the following function: 𝑓(𝑧)=𝑧+1−𝛼−𝜅1−𝛽2𝜅+(𝑒2−𝛽2−𝛼)^𝑖𝜃2𝑧². (43)

Theorem 15. Let the function 𝑓(𝑧) defined by (1) be in the class 𝜅−𝑉UCV(𝛼,𝛽). Then 1−1−𝛼−𝜅1−𝛽𝜅+()|𝑧|≤𝑓2−𝛽2−𝛼_′(𝑧)≤1+1−𝛼−𝜅1−𝛽𝜅+()|𝑧|.2−𝛽2−𝛼 (44) The result is sharp for the function given by (43).

4. Extreme Points

Theorem 16. Let the function 𝑓(𝑧) defined by (1) be in the class 𝜅−𝑉ST(𝛼,𝛽), with arg𝑎_𝑛=𝜃_𝑛, where 𝜃_𝑛+(𝑛−1)𝛿≡𝜋(mod2𝜋). Define 𝑓₁(𝑧)𝑓=𝑧,_𝑛(𝑧)=𝑧+1−𝛼−𝜅1−𝛽𝜅+()𝑒𝑛−𝛽𝑛−𝛼^𝑖𝜃𝑛𝑧^𝑛(𝑛≥2;𝑧∈𝑈). (45) Then 𝑓(𝑧)∈𝜅−𝑉ST(𝛼,𝛽) if and only if 𝑓(𝑧) can be expressed in the form 𝑓(𝑧)=∑^∞_𝑛=1𝜇_𝑛𝑓_𝑛(𝑧), where 𝜇_𝑛≥0 and ∑^∞_𝑛=1𝜇_𝑛=1.

Proof. If 𝑓(𝑧)=∑^∞_𝑛=1𝜇_𝑛𝑓_𝑛(𝑧) with 𝜇_𝑛≥0 and ∑^∞_𝑛=1𝜇_𝑛=1, then 𝑓(𝑧)=^∞_𝑛=1𝜇_𝑛𝑓_𝑛(𝑧)=𝜇₁𝑓₁(𝑧)+^∞_𝑛=2𝜇_𝑛𝑓_𝑛(𝑧)=1−^∞_𝑛=2𝜇_𝑛𝑧+^∞_𝑛=2𝜇_𝑛𝑧+1−𝛼−𝜅1−𝛽𝜅+()𝑒𝑛−𝛽𝑛−𝛼^𝑖𝜃𝑛𝑧^𝑛=𝑧+^∞_𝑛=21−𝛼−𝜅1−𝛽𝜅+()𝜇𝑛−𝛽𝑛−𝛼^𝑛𝑒^𝑖𝜃𝑛𝑧^𝑛. (46) But according to (17), we can see that ^∞_𝑛=2𝜅𝑛−𝛽+(𝑛−𝛼)1−𝛼−𝜅1−𝛽𝜅+()𝜇𝑛−𝛽𝑛−𝛼^𝑛𝑒^𝑖𝜃𝑛=^∞_𝑛=2𝜅𝑛−𝛽+(𝑛−𝛼)1−𝛼−𝜅1−𝛽𝜅+()𝜇𝑛−𝛽𝑛−𝛼^𝑛=^∞_𝑛=21−𝛼−𝜅1−𝛽𝜇_𝑛=1−𝜇₁1−𝛼−𝜅1−𝛽≤1−𝛼−𝜅1−𝛽. (47) Then 𝑓(𝑧) satisfies (17), hence 𝑓(𝑧)∈𝜅−𝑉ST(𝛼,𝛽).
Conversely, let the function 𝑓(𝑧) defined by (1) be in the class 𝜅−𝑉ST(𝛼,𝛽), and define 𝜇_𝑛=𝜅𝑛−𝛽+(𝑛−𝛼)𝑎1−𝛼−𝜅1−𝛽^𝑛𝜇,𝑛≥2,₁=1−^∞_𝑛=2𝜇_𝑛. (48) From Theorem 6, ∑^∞_𝑛=2𝜇_𝑛≤1 and so 𝜇₁≥0. Since 𝜇_𝑛𝑓_𝑛(𝑧)=𝜇_𝑛𝑧+𝑎_𝑛𝑧^𝑛, then ^∞_𝑛=1𝜇_𝑛𝑓_𝑛(𝑧)=𝑧+^∞_𝑛=2𝑎_𝑛𝑧^𝑛=𝑓(𝑧). (49) This completes the proof of Theorem 16.

Finally using the same technique used in Theorem 16 we get the following theorem.

Theorem 17. Let the function 𝑓(𝑧) defined by (1) be in the class 𝜅−𝑉UCV(𝛼,𝛽), with arg𝑎_𝑛=𝜃_𝑛, where 𝜃_𝑛+(𝑛−1)𝛿≡𝜋 (mod2𝜋). Define 𝑓₁(𝑧)𝑓=𝑧,_𝑛(𝑧)=𝑧+1−𝛼−𝜅1−𝛽𝑛𝜅+(𝑒𝑛−𝛽𝑛−𝛼)^𝑖𝜃𝑛𝑧^𝑛(𝑛≥2;𝑧∈𝑈). (50) Then 𝑓(𝑧)∈𝜅−𝑉UCV(𝛼,𝛽) if and only if 𝑓(𝑧) can be expressed in the form 𝑓(𝑧)=∑^∞_𝑛=1𝜇_𝑛𝑓_𝑛(𝑧), where 𝜇_𝑛≥0 and ∑^∞_𝑛=1𝜇_𝑛=1.

Acknowledgment

The authors thank the referees for their valuable suggestions which led to the improvement of this paper.