Hindawi Journal of Function Spaces Volume 2018, Article ID 6049512, 7 pages https://doi.org/10.1155/2018/6049512
Research Article Uniformly Alpha-Quasi-Convex Functions Defined by Janowski Functions Shahid Mahmood ,1 Sarfraz Nawaz Malik ,2 Sumbal Farman,2 S. M. Jawwad Riaz,2 and Shabieh Farwa 2 1
Department of Mechanical Engineering, Sarhad University of Science and Information Technology, Ring Road, Peshawar, Pakistan Department of Mathematics, COMSATS Institute of Information Technology, Wah Cantt, Pakistan
2
Correspondence should be addressed to Sarfraz Nawaz Malik;
[email protected] Received 25 October 2017; Revised 30 November 2017; Accepted 18 December 2017; Published 2 January 2018 Academic Editor: Hugo Leiva Copyright © 2018 Shahid Mahmood et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this work, we aim to introduce and study a new subclass of analytic functions related to the oval and petal type domain. This includes various interesting properties such as integral representation, sufficiency criteria, inclusion results, and the convolution properties for newly introduced class.
where 𝑤(0) = 0, |𝑤(𝑧)| < 1 for 𝑧 ∈ U. The class P[𝐴, 𝐵] consists of functions 𝑓(𝑧) which are analytic in U with 𝑓(0) = 1 and
1. Introduction and Preliminaries Let A be the class of functions 𝑓(𝑧) of the form ∞
𝑓 (𝑧) = 𝑧 + ∑ 𝑎𝑛 𝑧𝑛 ,
𝑓 (𝑧) ≺
(1)
𝑛=2
which are analytic in the open unit disk U = {𝑧 : |𝑧| < 1}. Let S, S∗ (𝛾), C(𝛾), and K(𝛾) be the subclasses of A consisting of functions which are univalent, starlike of order 𝛾, convex of order 𝛾, and close-to-convex of order 𝛾, respectively, 0 ≤ 𝛾 < 1. Clearly S∗ (0) = S∗ , the class of starlike univalent functions, C(0) = C, the class of convex univalent functions, and K(0) = K, the class of close-to-convex univalent functions. 𝑛 For analytic functions 𝑓(𝑧) = 𝑧+∑∞ 𝑛=2 𝑎𝑛 𝑧 and 𝑔(𝑧) = 𝑧+ ∞ ∑𝑛=2 𝑏𝑛 𝑧𝑛 , the convolution or Hadamard product is denoted by 𝑓 ∗ 𝑔 and is defined by ∞
(𝑓 ∗ 𝑔) (𝑧) = 𝑧 + ∑ 𝑎𝑛 𝑏𝑛 𝑧𝑛 .
(2)
𝑛=2
A function 𝑓(𝑧) is said to be subordinate to the function 𝑔(𝑧), written symbolically as 𝑓(𝑧) ≺ 𝑔(𝑧), if there exists a Schwarz function 𝑤(𝑧) such that 𝑓 (𝑧) = 𝑔 (𝑤 (𝑧)) , 𝑧 ∈ U,
(3)
1 + 𝐴𝑧 , 1 + 𝐵𝑧
(4)
where −1 ≤ 𝐵 < 𝐴 ≤ 1. If a function 𝑓(𝑧) belongs to P[𝐴, 𝐵], then it maps U on to the domain Ω[𝐴, 𝐵], defined by 1 − 𝐴𝐵 𝐴 − 𝐵 < }. Ω [𝐴, 𝐵] = {𝑤 : 𝑤 − 1 − 𝐵2 1 − 𝐵2
(5)
The domain Ω[𝐴, 𝐵] represents an open circular disk, having center at real axis and whose diameter points are 𝑑1 = (1 − 𝐴)/(1 − 𝐵) and 𝑑2 = (1 + 𝐴)/(1 + 𝐵). For 𝐵 = −1 and 𝐴 = 1, the class P[𝐴, 𝐵] reduces to the well-known class P of Carath`eodory functions; see [1]. The class P[𝐴, 𝐵] is connected with the class P of Carath`eodory functions by the following relation. 𝑓 (𝑧) ∈ P ⇐⇒
(𝐴 + 1) 𝑓 (𝑧) − (𝐴 − 1) ∈ P [𝐴, 𝐵] . (𝐵 + 1) 𝑓 (𝑧) − (𝐵 − 1)
(6)
Janowski [2] introduced the classes S∗ [𝐴, 𝐵] and C[𝐴, 𝐵] and studied them comprehensively. He established the Alexander type relation between these classes.
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Journal of Function Spaces
In 1999, Kanas and Wi´sniowska [3] gave the concept of general conic domain by introducing a domain Ω𝛽 , defined by Ω𝛽 = {𝑢 + 𝑖V : 𝑢 > 𝛽√(𝑢 − 1)2 + V2 } .
(7)
The domain Ω𝛽 represents a right half plane when 𝛽 = 0, the hyperbolic regions for 0 < 𝛽 < 1, a parabolic region for 𝛽 = 1, and it represents elliptic regions when 𝛽 > 1. The extremal function for conic domain Ω𝛽 is the mapping 𝑝𝛽 (𝑧) with 𝑝𝛽 (0) = 1 and 𝑝𝛽 (0) > 0 and is defined by
1+𝑧 { , { { 1−𝑧 { { { 2 { 1 + √𝑧 2 { { { ) , 1 + 2 (log { { 𝜋 1 − √𝑧 { { 𝑝𝛽 (𝑧) = {1 + 2 sin ℎ2 [( 2 arccos 𝛽) arctan ℎ√𝑧] , { { 1 − 𝛽2 𝜋 { { { { { 𝑢(𝑧)/𝑡 { 1 1 1 𝜋 { { 1 + 𝑑𝑥) + 2 sin ( , ∫ { { 2 { 𝛽 −1 2𝑋 (𝑡) 0 𝛽 −1 √1 − 𝑥2 √1 − (𝑡𝑥)2 { where 𝑢(𝑧) = (𝑧 − √𝑡)/(1 − √𝑡𝑧), 𝑡 ∈ (0, 1), 𝑧 ∈ U, and 𝑧 is chosen such that 𝛽 = cosh (𝜋𝑋 (𝑡)/4𝑋(𝑡)), 𝑋(𝑡) is Legendre’s complete elliptic integral of first kind, and 𝑋 (𝑡) is complementary integral of 𝑋(𝑡); for more details, see [3]. Meanwhile, Kanas and Wi´sniowska [3, 4] introduced a class P(𝑝𝛽 ) which maps U on to the conic regions, defined by the domain Ω𝛽 . They also defined the classes 𝛽 − UCV and 𝛽 − ST of 𝛽-uniformly convex functions and their corresponding 𝛽-starlike functions; see [3, 4]. Recently Noor and Malik [5] introduced new geometrical structures of oval and petal type shape as image domain and defined the class 𝛽 − P[𝐴, 𝐵] of functions which give these types of mappings. The concepts of Janowski functions and conic domains are combined together to define the class 𝛽 − P[𝐴, 𝐵] which represents the oval and petal type regions as image domain. Definition 1 (see [5]). A function 𝑝(𝑧) ∈ 𝛽 − P[𝐴, 𝐵] if and only if, 𝑝 (𝑧) ≺ 𝑝𝛽 (𝐴, 𝐵; 𝑧) ,
(𝛽 ≥ 0, −1 ≤ 𝐵 < 𝐴 ≤ 1) ,
(9)
where 𝑝𝛽 (𝐴, 𝐵; 𝑧) =
(𝐴 + 1) 𝑝𝛽 (𝑧) − (𝐴 − 1) (𝐵 + 1) 𝑝𝛽 (𝑧) − (𝐵 − 1)
𝛽 = 1, (8)
0 < 𝛽 < 1, 𝛽 > 1,
or, equivalently, Ω𝛽 [𝐴, 𝐵] = {𝑢 + 𝑖V : [(𝐵2 − 1) (𝑢2 + V2 ) − 2 (𝐴𝐵 − 1) 𝑢 + (𝐴2 + 1)] 2
2
(12)
2
> [−2 (𝐵 + 1) (𝑢 + V ) + 2 (𝐴 + 𝐵 + 2) 𝑢 − 2 (𝐴 + 1)2 ]} . Noor and Malik [5] also introduced two new classes 𝛽 − UCV[𝐴, 𝐵] and 𝛽 − ST[𝐴, 𝐵] related to the domain Ω𝛽 [𝐴, 𝐵] whose geometry is oval and petal type regions subject to the values of 𝐴, 𝐵, and beta. These classes are defined as follows. Definition 2 (see [5]). A function 𝑓(𝑧) ∈ A is said to be in the class 𝛽 − UCV[𝐴, 𝐵], 𝑘 ≥ 0, −1 ≤ 𝐵 < 𝐴 ≤ 1, if and only if
(10)
with 𝑝𝛽 (𝑧) as defined in (8). Graphically, the function 𝑝(𝑧) ∈ 𝛽 − P[𝐴, 𝐵] takes all the values in the domain Ω𝛽 [𝐴, 𝐵], −1 ≤ 𝐵 < 𝐴 ≤ 1, which is defined as (𝐵 − 1) 𝑤 (𝑧) − (𝐴 − 1) ) (𝐵 + 1) 𝑤 (𝑧) − (𝐴 + 1) (𝐵 − 1) 𝑤 (𝑧) − (𝐴 − 1) − 1} > 𝛽 (𝐵 + 1) 𝑤 (𝑧) − (𝐴 + 1)
𝛽 = 0,
Re (
(𝐵 − 1) ((𝑧𝑓 (𝑧)) /𝑓 (𝑧)) − (𝐴 − 1)
(𝐵 + 1) ((𝑧𝑓 (𝑧)) /𝑓 (𝑧)) − (𝐴 + 1)
)
(𝐵 − 1) ((𝑧𝑓 (𝑧)) /𝑓 (𝑧)) − (𝐴 − 1) , − 1 > 𝛽 (𝐵 + 1) ((𝑧𝑓 (𝑧)) /𝑓 (𝑧)) − (𝐴 + 1)
(13)
or, equivalently,
Ω𝛽 [𝐴, 𝐵] = {𝑤 (𝑧) : Re (
(11)
(𝑧𝑓 (𝑧)) 𝑓 (𝑧)
∈ 𝛽 − P [𝐴, 𝐵] .
(14)
Journal of Function Spaces
3
Definition 3 (see [5]). A function 𝑓(𝑧) ∈ A is said to be in the class 𝛽 − ST[𝐴, 𝐵], 𝑘 ≥ 0, −1 ≤ 𝐵 < 𝐴 ≤ 1, if and only if Re (
(𝐵 − 1) (𝑧𝑓 (𝑧) /𝑓 (𝑧)) − (𝐴 − 1) (𝐵 + 1) (𝑧𝑓 (𝑧) /𝑓 (𝑧)) − (𝐴 + 1)
)
(𝐵 − 1) (𝑧𝑓 (𝑧) /𝑓 (𝑧)) − (𝐴 − 1) , > 𝛽 − 1 (𝐵 + 1) (𝑧𝑓 (𝑧) /𝑓 (𝑧)) − (𝐴 + 1)
(15)
(16)
Throughout this paper, we assume that 𝛽 ≥ 0, −1 ≤ 𝐵 < 𝐴 ≤ 1, −1 ≤ 𝐷 < 𝐶 ≤ 1, and 0 ≤ 𝛼 ≤ 1 unless otherwise stated.
2. A Set of Lemmas
Motivated from the recent work presented by Noor and Malik [5], we define the following new subclass of analytic functions associated with domain Ω𝛽 [𝐴, 𝐵]. Definition 4. A function 𝑓(𝑧) ∈ A is said to be in the class 𝛽 − UQ𝛼 [𝐴, 𝐵, 𝐶, 𝐷], 𝛽 ≥ 0, 0 ≤ 𝛼 ≤ 1, −1 ≤ 𝐵 < 𝐴 ≤ 1, −1 ≤ 𝐷 < 𝐶 ≤ 1, if and only if (𝐵 − 1) 𝑤 (𝑧) − (𝐴 − 1) ) (𝐵 + 1) 𝑤 (𝑧) − (𝐴 + 1) (𝐵 − 1) 𝑤 (𝑧) − (𝐴 − 1) > 𝛽 − 1 , (𝐵 + 1) 𝑤 (𝑧) − (𝐴 + 1)
𝑐𝑛 ≤ 𝛿 (𝐴, 𝐵, 𝛽) , (𝐴 − 𝐵) 𝛿𝛽
𝛿 (𝐴, 𝐵, 𝛽) =
(18)
8 (cos−1 𝛽) { { { , { { 𝜋2 (1 − 𝛽2 ) { { {8 𝛿𝛽 = { , { 𝜋2 { { { { 𝜋2 { { , { 4√𝑡 (𝛽2 − 1) 𝑅2 (𝑡) (1 + 𝑡)
2
(19)
(22)
0 ≤ 𝛽 < 1, 𝛽 = 1,
(23)
𝛽 > 1.
∞
It follows from the above definition that 𝑓(𝑧) ∈ 𝛽 − UQ𝛼 [𝐴, 𝐵, 𝐶, 𝐷] if and only if
∈ 𝛽 − UK [𝐴, 𝐵, 𝐶, 𝐷] ,
,
Lemma 6 (see [5]). Let 𝑔(𝑧) ∈ 𝛽 − CV[𝐶, 𝐷] with 𝛽 ≥ 0 and be given by
for some 𝑔(𝑧) ∈ 𝛽 − UCV[𝐶, 𝐷].
𝐹 (𝑧) = (1 − 𝛼) 𝑓 (𝑧) + 𝛼𝑧𝑓 (𝑧)
(21)
2
where (𝑧𝑓 (𝑧)) 𝑓 (𝑧) + 𝛼 𝑔 (𝑧) 𝑔 (𝑧)
𝑛 Lemma 5 (see [5]). Let 𝑝(𝑧) = 1 + ∑∞ 𝑛=1 𝑐𝑛 𝑧 ∈ 𝛽 − P[𝐴, 𝐵]. Then
(17)
or 𝑤 (𝑧) ∈ 𝛽 − P [𝐴, 𝐵] ,
To prove our main results we need the following Lemmas.
where
Re (
𝑤 (𝑧) = (1 − 𝛼)
(3) 0 − UQ𝛼 [1, −1, 1, −1] = Q𝛼 is the well known class of alpha-quasi-convex functions, introduced and studied by Noor and Aboudi [8]. (4) 0 − UQ𝛼 [𝐴, 𝐵, 1, −1] = Q𝛼 [𝐴, 𝐵] is the well known class of alpha Janowski convex functions, introduced and studied by Selvaraj and Thirupathi [9].
or, equivalently, 𝑧𝑓 (𝑧) ∈ 𝛽 − P [𝐴, 𝐵] . 𝑓 (𝑧)
(2) 𝛽 − UQ𝛼 [1, −1, 1, −1] = 𝛽 − UK is the well known class of uniformly close-to-convex functions, introduced and studied by Noor et al. [7].
𝑔 (𝑧) = 𝑧 + ∑ 𝑏𝑛 𝑧𝑛 .
(24)
𝑛−2 1 (𝐶 − 𝐷) 𝛿𝛽 − 2𝑗𝐷 , 𝑏𝑛 ≤ ∏ 𝑛 𝑗=0 2 (𝑗 + 1)
(25)
𝑛=2
Then (20)
where 𝛽 − UK[𝐴, 𝐵, 𝐶, 𝐷] is the class of uniformly Janowski close-to-convex functions, introduced and studied by Mahmood et al. [6]. Special Cases. By setting certain values to different parameters, the class 𝛽 − UQ𝛼 [𝐴, 𝐵, 𝐶, 𝐷] coincides with several renowned classes of analytic functions, as given below. (1) 𝛽 − UQ0 [𝐴, 𝐵, 𝐶, 𝐷] = 𝛽 − UK[𝐴, 𝐵, 𝐶, 𝐷], 𝛽 − UQ1 [𝐴, 𝐵, 𝐶, 𝐷] = 𝛽 − UQ[𝐴, 𝐵, 𝐶, 𝐷] are well-known classes of uniformly Janowski close-toconvex and uniformly Janowski quasi-convex functions respectively, introduced and studied by Mahmood et al. [6].
where 𝛿𝛽 is defined by (23). Lemma 7 (see [10]). Let 𝑓(𝑧) and 𝑔(𝑧) be in the classes C and S∗ , respectively. Then for every function 𝐹(𝑧) analytic in U with 𝐹(0) = 1, we have 𝑓 (𝑧) ∗ 𝑔 (𝑧) 𝐹 (𝑧) ∈ co (𝐹 (U)) , 𝑧 ∈ U, 𝑓 (𝑧) ∗ 𝑔 (𝑧)
(26)
where co𝐹(U) denotes the closed convex hull 𝐹(U). Lemma 8 (see [6]). Let 𝑓(𝑧) ∈ 𝛽 − UK[𝐴, 𝐵, 𝐶, 𝐷] and 𝜑(𝑧) ∈ C. Then (𝜑 ∗ 𝑓)(𝑧) ∈ 𝛽 − UK[𝐴, 𝐵, 𝐶, 𝐷].
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Journal of Function Spaces ∞
3. Main Results
+ ∑ 𝑛 (1 + 𝛼 (𝑛 − 1)) 𝑎𝑛 )
Theorem 9. Let 𝑓(𝑧) ∈ 𝛽 − UQ𝛼 [𝐴, 𝐵, 𝐶, 𝐷]. Then, the function 𝐹(𝑧) is defined such that 𝑓 (𝑧) =
𝑧
1 1−1/𝛼 𝑧 ∫ 𝑡2−1/𝛼 𝐹 (𝑡) 𝑑𝑡, 𝛼 0
(𝛼 > 0, 𝑧 ∈ U)
𝑛=2 ∞
+ ∑ |𝐵 + 1| 𝑛 (1 + 𝛼 (𝑛 − 1)) 𝑎𝑛
(27)
𝑛=2
∞ 𝑛−2 𝛿 𝛽
belongs to the class 𝛽 − UK[𝐴, 𝐵, 𝐶, 𝐷].
+ ∑∏ 𝑛=2 𝑗=0
Proof. From (27), we can write
(1 − 𝛼) 𝑓 (𝑧) + 𝛼𝑧𝑓 (𝑧) = 𝐹 (𝑧) ,
2 (𝑗 + 1)
|𝐴 + 1|] < |𝐵 − 𝐴| . ] (29)
(𝑧 ∈ U)
(28)
and the result follows by using (20). Theorem 10. A function 𝑓(𝑧) having the form (1) is in the class 𝛽 − UQ𝛼 [𝐴, 𝐵, 𝐶, 𝐷] if it satisfies ∞ 𝑛−2 𝛿 (𝐶 − 𝐷) − 2𝑗𝐷 [2 (𝛽 + 1) ( ∑ ∏ 𝛽 2 (𝑗 + 1) 𝑛=2 𝑗=0 [
(𝐶 − 𝐷) − 2𝑗𝐷
Proof. Assuming that (29) holds true, then it is sufficient to show that (𝐵 − 1) 𝑤 (𝑧) − (𝐴 − 1) − 1 𝛽 (𝐵 + 1) 𝑤 (𝑧) − (𝐴 + 1) (30) (𝐵 − 1) 𝑤 (𝑧) − (𝐴 − 1) − 1] < 1. − Re [ (𝐵 + 1) 𝑤 (𝑧) − (𝐴 + 1) Now consider
(𝐵 − 1) ((1 − 𝛼) (𝑓 (𝑧) /𝑔 (𝑧)) + 𝛼 ((𝑧𝑓 (𝑧)) /𝑔 (𝑧))) − (𝐴 − 1) − 1 𝛽 (𝐵 + 1) ((1 − 𝛼) (𝑓 (𝑧) /𝑔 (𝑧)) + 𝛼 ((𝑧𝑓 (𝑧)) /𝑔 (𝑧))) − (𝐴 + 1)
(𝐵 − 1) ((1 − 𝛼) (𝑓 (𝑧) /𝑔 ) + 𝛼 ((𝑧𝑓 (𝑧)) /𝑔 )) − (𝐴 − 1)
− 1] (𝐵 + 1) ((1 − 𝛼) (𝑓 (𝑧) /𝑔 (𝑧)) + 𝛼 ((𝑧𝑓 (𝑧)) /𝑔 (𝑧))) − (𝐴 + 1) ] [ (𝐵 − 1) [𝑓 (𝑧) /𝑔 (𝑧) + 𝛼 (𝑧𝑓 (𝑧) /𝑔 (𝑧))] − (𝐴 − 1) − 1 ≤ (𝛽 + 1) (𝐵 + 1) [𝑓 (𝑧) /𝑔 (𝑧) + 𝛼 (𝑧𝑓 (𝑧) /𝑔 (𝑧))] − (𝐴 + 1) (𝐵 − 1) [𝑓 (𝑧) + 𝛼𝑧𝑓 (𝑧)] − (𝐴 − 1) 𝑔 (𝑧) = (𝛽 + 1) − 1 (𝐵 + 1) [𝑓 (𝑧) + 𝛼𝑧𝑓 (𝑧)] − (𝐴 + 1) 𝑔 (𝑧) (𝐵 − 1) [𝑓 (𝑧) + 𝛼𝑧𝑓 (𝑧)] − (𝐴 − 1) 𝑔 (𝑧) − (𝐵 + 1) [𝑓 (𝑧) + 𝛼𝑧𝑓 (𝑧)] + (𝐴 + 1) 𝑔 (𝑧) = (𝛽 + 1) (𝐵 + 1) [𝑓 (𝑧) + 𝛼𝑧𝑓 (𝑧)] − (𝐴 + 1) 𝑔 (𝑧) 𝑔 (𝑧) − 𝑓 (𝑧) − 𝛼𝑧𝑓 (𝑧) = 2 (𝛽 + 1) (𝐵 + 1) [𝑓 (𝑧) + 𝛼𝑧𝑓 (𝑧)] − (𝐴 + 1) 𝑔 (𝑧) 𝑛−1 𝑛−1 𝑛−1 − 1 − ∑∞ − 𝛼 ∑∞ 1 + ∑∞ 𝑛=2 𝑛𝑏𝑛 𝑧 𝑛=2 𝑛𝑎𝑛 𝑧 𝑛=2 𝑛 (𝑛 − 1) 𝑎𝑛 𝑧 ≤ 2 (𝛽 + 1) ∞ ∞ ∞ (𝐵 + 1) (1 + ∑𝑛=2 𝑛𝑎𝑛 𝑧𝑛−1 − 𝛼 ∑𝑛=2 𝑛 (𝑛 − 1) 𝑎𝑛 𝑧𝑛−1 ) − (𝐴 + 1) (1 + ∑𝑛=2 𝑛𝑏𝑛 𝑧𝑛−1 ) 𝑛−1 𝑛−1 ∑∞ − ∑∞ 𝑛=2 𝑛𝑏𝑛 𝑧 𝑛=2 𝑛 (1 + 𝛼 (𝑛 − 1)) 𝑎𝑛 𝑧 = 2 (𝛽 + 1) ∞ ∞ (𝐵 − 𝐴) + (𝐵 + 1) (∑𝑛=2 𝑛 (1 + 𝛼𝑛 − 𝛼) 𝑎𝑛 𝑧𝑛−1 ) − (𝐴 + 1) (∑𝑛=2 𝑛𝑏𝑛 𝑧𝑛−1 ) ∞ ∑∞ 𝑛=2 𝑛 𝑏𝑛 + ∑𝑛=2 𝑛 (1 + 𝛼 (𝑛 − 1)) 𝑎𝑛 ≤ 2 (𝛽 + 1) ∞ ∞ |𝐵 − 𝐴| − (𝐵 + 1) (∑𝑛=2 𝑛 (1 + 𝛼𝑛 − 𝛼) 𝑎𝑛 𝑧𝑛−1 ) − (𝐴 + 1) (∑𝑛=2 𝑛𝑏𝑛 𝑧𝑛−1 ) ∞ ∑∞ 𝑛=2 𝑛 𝑏𝑛 + ∑𝑛=2 𝑛 (1 + 𝛼 (𝑛 − 1)) 𝑎𝑛 ≤ 2 (𝛽 + 1) ∞ ∞ |𝐵 − 𝐴| − |𝐵 + 1| ∑𝑛=2 𝑛 (1 + 𝛼 (𝑛 − 1)) 𝑎𝑛 − |𝐴 + 1| ∑𝑛=2 𝑛 𝑏𝑛 𝑛−2 ∞ ∑∞ 𝑛=2 ∏𝑗=0 (𝛿𝛽 (𝐶 − 𝐷) − 2𝑗𝐷 /2 (𝑗 + 1)) + ∑𝑛=2 𝑛 (1 + 𝛼 (𝑛 − 1)) 𝑎𝑛 . ≤ 2 (𝛽 + 1) ∞ ∞ 𝑛−2 |𝐵 − 𝐴| − |𝐵 + 1| ∑𝑛=2 𝑛 (1 + 𝛼𝑛 − 𝛼) 𝑎𝑛 − |𝐴 + 1| ∑𝑛=2 ∏𝑗=0 (𝛿𝛽 (𝐶 − 𝐷) − 2𝑗𝐷 /2 (𝑗 + 1))
− Re [
(31)
Journal of Function Spaces
5
In the last inequality, we have used the following coefficient inequality for functions belonging to the class 𝛽 − CV[𝐶, 𝐷] (see [5]): 𝑛−2 1 𝛿𝛽 (𝐶 − 𝐷) − 2𝑗𝐷 . 𝑏𝑛 ≤ ∏ 𝑛 𝑗=0 2 (𝑗 + 1)
(32)
The last inequality is bounded above by 1, if
∞ 𝑛−2 𝛿 𝛽
[2 (𝛽 + 1) ( ∑ ∏ 𝑛=2 𝑗=0
[
(𝐶 − 𝐷) − 2𝑗𝐷
Theorem 13. Let 𝑓(𝑧) ∈ 𝛽 − UQ𝛼 [𝐴, 𝐵, 𝐶, 𝐷] and be of the form (1). Then for 𝑛 ≥ 2, we have 𝑛−2 𝛿 (𝐶 − 𝐷) − 2𝑗𝐷 𝛽 1 [ ∏ 𝑎𝑛 ≤ [𝑛 + 𝛼𝑛 (𝑛 − 1)] 𝑗=0 2 (𝑗 + 1) [ 𝑛−1 𝑗−2 𝛿 (𝐶 − 𝐷) − 2𝑚𝐷 ] 𝛽 + 𝛿 (𝐴, 𝐵, 𝛽) ∑ ∏ , 2 (𝑚 + 1) 𝑗=1 𝑚=0 ] where 𝛿(𝐴, 𝐵, 𝛽) is given by (22).
2 (𝑗 + 1)
Proof. By definition, for 𝑓(𝑧) ∈ 𝛽 − UQ𝛼 [𝐴, 𝐵, 𝐶, 𝐷], we get
∞
(𝑧𝑓 (𝑧)) 𝑓 (𝑧) +𝛼 = 𝑝 (𝑧) , (1 − 𝛼) 𝑔 (𝑧) 𝑔 (𝑧)
+ ∑ 𝑛 (1 + 𝛼 (𝑛 − 1)) 𝑎𝑛 ) 𝑛=2
(33)
∞
∞
𝑛=2
+ ∑∏ 𝑛=2 𝑗=0
(𝐶 − 𝐷) − 2𝑗𝐷 2 (𝑗 + 1)
𝑝 (𝑧) = 1 + ∑ 𝑐𝑛 𝑧𝑛 ≺ 𝑝𝛽 (𝐴, 𝐵; 𝑧) .
|𝐴 + 1|] < |𝐵 − 𝐴| . ]
We can write (37) as
[2 (𝛽 + 1) ( ∑ ∏ 𝑛=2 𝑗=0
[
(𝐶 − 𝐷) − 2𝑗𝐷 2 (𝑗 + 1)
Now using the series expansion, (39) reduces to ∞
𝑛=2
+ ∑ 𝑛 𝑎𝑛 )
𝑛=2
∞
𝑛=2
(34)
+ ∑∏ 𝑛=2 𝑗=0
).
𝑛=2
(41)
𝑗=1
This implies that
When 𝛼 = 1, Theorem 10 takes the form of following known result, proved in [6]. Corollary 12. A function 𝑓(𝑧) ∈ A is said to be in the class 𝛽 − UQ[𝐴, 𝐵, 𝐶, 𝐷] if it satisfies the condition ∞ ∞ 𝑛−2 𝛿 (𝐶 − 𝐷) − 2𝑗𝐷 [2 (𝛽 + 1) ( ∑ ∏ 𝛽 + ∑ 𝑛2 𝑎𝑛 ) 2 (𝑗 + 1) 𝑛=2 𝑗=0 𝑛=2 [ ∞
∞ 𝑛−2 𝛿 (𝐶 − 𝐷) − 2𝑗𝐷 𝛽 + ∑∏ |𝐴 + 1|] < |𝐵 − 𝐴| . 2 (𝑗 + 1) 𝑛=2 𝑗=0 ]
) (1 + ∑ 𝑐𝑛−1 𝑧
𝑛𝑎𝑛 + 𝛼𝑛 (𝑛 − 1) 𝑎𝑛 = 𝑛𝑏𝑛 + ∑ 𝑗𝑏𝑗 𝑐𝑛−𝑗 .
then 𝑓(𝑧) ∈ 𝛽 − UK[𝐴, 𝐵, 𝐶, 𝐷].
𝑛=2
(40) 𝑛−1
𝑛−1
(𝐶 − 𝐷) − 2𝑗𝐷 |𝐴 + 1|] < |𝐵 − 𝐴| , 2 (𝑗 + 1) ]
+ ∑ |𝐵 + 1| 𝑛2 𝑎𝑛
∞
Equating coefficients of 𝑧𝑛−1 on both sides, one may get
𝑛=2
𝑛−1
= (1 + ∑ 𝑛𝑏𝑛 𝑧
𝑛=2
∞
∞ 𝑛−2 𝛿 𝛽
(39)
1 + ∑ 𝑛𝑎𝑛 𝑧𝑛−1 + 𝛼 ∑ 𝑛 (𝑛 − 1) 𝑎𝑛 𝑧𝑛−1
∞
+ ∑ |𝐵 + 1| 𝑛 𝑎𝑛
𝑓 (𝑧) − 𝛼𝑧𝑓 (𝑧) = 𝑝 (𝑧) 𝑔 (𝑧) .
∞
Corollary 11. If a function 𝑓(𝑧) ∈ A satisfies
(38)
𝑛=1
When 𝛼 = 0, Theorem 10 reduces to the following known result, proved in [6].
∞ 𝑛−2 𝛿 𝛽
(37)
where
+ ∑ |𝐵 + 1| 𝑛 (1 + 𝛼 (𝑛 − 1)) 𝑎𝑛 ∞ 𝑛−2 𝛿 𝛽
(36)
(35)
𝑛−1 𝑎𝑛 [𝑛 + 𝛼𝑛 (𝑛 − 1)] ≤ 𝑛 𝑏𝑛 + ∑ 𝑗 𝑏𝑗 𝑐𝑛−𝑗 .
(42)
𝑗=1
Now applying Lemmas 5 and 6, (42) reduces to 𝑛−2 𝛿 (𝐶 − 𝐷) − 2𝑗𝐷 𝛽 1 [ ∏ 𝑎𝑛 ≤ [𝑛 + 𝛼𝑛 (𝑛 − 1)] 𝑗=0 2 (𝑗 + 1) [ 𝑛−1 𝑗−2 𝛿 (𝐶 − 𝐷) − 2𝑚𝐷 ] 𝛽 + 𝛿 (𝐴, 𝐵, 𝛽) ∑ ∏ , 2 + 1) (𝑚 𝑗=1 𝑚=0 ]
(43)
where 𝛿(𝐴, 𝐵, 𝛽) is given by (22). This completes the proof. By setting 𝛼 = 0 in Theorem 13, we have the following known result, proved by Mahmood et al. [6].
6
Journal of Function Spaces
Corollary 14. Let 𝑓(𝑧) ∈ 𝛽 − UK[𝐴, 𝐵, 𝐶, 𝐷] and 𝑎𝑛 be the 𝑛th coefficient of Taylor series of 𝑓(𝑧). Then, for 𝑛 ≥ 2, we obtain
𝑛−2 𝛿 𝛽
(𝐶 − 𝐷) − 2𝑗𝐷
1 [ + 𝛿 (𝐴, 𝐵, 𝛽) ∏ 𝑎𝑛 ≤ 𝑛 𝑗=0 2 (𝑗 + 1) [ 𝑛−1 𝑗−2 𝛿 (𝐶 − 𝐷) − 2𝑚𝐷 |𝐴 − 𝐵| 𝛿 𝛽 𝛽 ], ⋅ ∑∏ 2 + 1) (𝑚 𝑚=0 𝑗=1 ]
(44)
Theorem 19. Let 𝑓(𝑧) ∈ 𝛽 − UQ𝛼 [𝐴, 𝐵, 𝐶, 𝐷], 𝛼 ≥ 0, and 𝜙(𝑧) be any convex univalent function in U. Then 𝑓 (𝑧) ∗ 𝜙 (𝑧) ∈ 𝛽 − UQ𝛼 [𝐴, 𝐵, 𝐶, 𝐷] ,
(1 − 𝛼) 𝑓 (𝑧) + 𝛼𝑧𝑓 (𝑧) ∈ 𝛽 − UK [𝐴, 𝐵, 𝐶, 𝐷] .
For 𝛼 = 1, Theorem 13 takes the form of following known result, proved by Mahmood et al. [6]. Corollary 15. Let the function 𝑓(𝑧) of the form (1) belong to the class 𝛽 − UQ[𝐴, 𝐵, 𝐶, 𝐷]. Then,
∈ 𝛽 − UK [𝐴, 𝐵, 𝐶, 𝐷] ,
(45)
((1 − 𝛼) (𝑓 ∗ 𝜙) (𝑧) + 𝛼𝑧 (𝑓 ∗ 𝜙) (𝑧))
Corollary 16. Let 𝑓(𝑧) ∈ 𝛽−UK and 𝑎𝑛 be the 𝑛th coefficient of Taylor series of 𝑓(𝑧). Then,
Hence by (20), 𝑓(𝑧) ∗ 𝜙(𝑧) ∈ 𝛽 − UQ𝛼 [𝐴, 𝐵, 𝐶, 𝐷]. This completes the proof.
Theorem 20. Let 𝑓(𝑧) ∈ 𝛽 − UQ𝛼 [𝐴, 𝐵, 𝐶, 𝐷]. Then the functions 𝜙𝑖 (𝑧), such that 𝑧
𝜙1 (𝑧) = ∫
0
𝜙2 (𝑧) =
𝑓 (𝑡) 𝑑𝑡, 𝑡
2 𝑧 ∫ 𝑓 (𝑡) 𝑑𝑡, 𝑧 0
(53)
𝑧
(46)
By Putting 𝛽 = 0, 𝐶 = 1, and 𝐷 = −1 in Theorem 13, we have the following known result proved by Selvaraj and Thirupathi [9]. Corollary 17. Let 𝑓(𝑧) ∈ Q𝛼 [𝐴, 𝐵] and be of the form (1). Then, 1 (𝐴 − 𝐵) (𝑛 − 1) [1 + ], 𝑎𝑛 ≤ 1 + 𝛼 (𝑛 − 1) 2
𝑓 (𝑡) − 𝑓 (𝑡) 𝜙3 (𝑧) = ∫ 𝑑𝑡, 𝑡 − 𝑡𝑥 0 𝜙4 (𝑧) =
|𝑥| ≤ 1, 𝑥 ≠ 1,
𝜆 + 1 𝑧 𝜆−1 ∫ 𝑡 𝑓 (𝑡) 𝑑𝑡, 𝑧𝜆 0
Re (𝜆) > 0 (𝑧 ∈ U) ,
belong to the class 𝛽 − UQ𝛼 [𝐴, 𝐵, 𝐶, 𝐷]. Proof. The proof follows immediately, when we observe that 𝜙𝑖 (𝑧) = (𝑓 ∗ 𝜓𝑖 ) (𝑧) , 𝑖 = 1, 2, 3, 4 (𝑧 ∈ U) ,
(54)
where 𝜓𝑖 (𝑧) is convex in U such that (47) (𝑛 ≥ 2) .
When we take 𝐴 = 𝐶 = 1, 𝐵 = 𝐷 = −1, and 𝛽 = 0 in Theorem 13, we obtain the following known result proved by Noor and Aboudi [8]. Corollary 18. Let 𝑓(𝑧) ∈ Q𝛼 [𝐴, 𝐵] and 𝑎𝑛 be the 𝑛th coefficient of Taylor series of 𝑓(𝑧). Then, (𝑛 ≥ 2) .
(52)
Applications of Theorem 19
When we consider 𝐴 = 𝐶 = 1, 𝐵 = 𝐷 = −1, and 𝛼 = 0 in Theorem 13, we obtain the following known result, proved by Noor et al. [7].
(𝑛 ≥ 2) .
(51)
which implies that
∈ 𝛽 − UK [𝐴, 𝐵, 𝐶, 𝐷] .
where 𝑛 ≥ 2 and 𝛿(𝐴, 𝐵, 𝛽) is given by (22).
𝑛−1 (𝛿𝛽 ) 𝑗−1 (𝛿𝛽 )𝑛−1 𝛿𝛽 , + ∑ 𝑎𝑛 ≤ 𝑛! 𝑛 𝑗=0 (𝑗 − 1)!
(50)
Using Lemma 8, one may have ((1 − 𝛼) 𝑓 (𝑧) + 𝛼𝑧𝑓 (𝑧)) ∗ 𝜙 (𝑧)
𝑛−2 1 [ 𝛿𝛽 (𝐶 − 𝐷) − 2𝑗𝐷 + 𝛿 (𝐴, 𝐵, 𝛽) 𝑎𝑛 ≤ 2 ∏ 𝑛 𝑗=0 2 (𝑗 + 1) [ 𝑛−1 𝑗−2 𝛿 (𝐶 − 𝐷) − 2𝑚𝐷 |𝐴 − 𝐵| 𝛿 𝛽 𝛽 ], ⋅ ∑∏ 2 + 1) (𝑚 𝑗=1 𝑚=0 ]
(49)
Proof. Let 𝑓(𝑧) ∈ 𝛽 − UQ𝛼 [𝐴, 𝐵, 𝐶, 𝐷], 𝛼 ≥ 0. Then by (20), we have
where 𝛿(𝐴, 𝐵, 𝛽) is given by (22).
𝑛 , 𝑎𝑛 ≤ 1 + 𝛼 (𝑛 − 1)
(𝑧 ∈ U) .
(48)
∞ 1 𝜓1 = − log (1 − 𝑧) = ∑ 𝑧𝑛 , 𝑛 𝑛=1
𝜓2 =
− [𝑧 + log (1 − 𝑧)] , 𝑧
1 1 − 𝑥𝑧 𝜓3 = log ( ), 1−𝑥 1−𝑧 ∞
1+𝜆 𝑛 𝑧 , 𝑛 𝑛=1 + 𝜆
𝜓4 = ∑
Re (𝜆) > 0, (𝑧 ∈ U) .
(55)
Journal of Function Spaces
7
Theorem 21. For 𝛼 ≥ 0, we have 𝛽 − UQ𝛼 [𝐴, 𝐵, 𝐶, 𝐷] ⊆ 𝛽 − UK[𝐴, 𝐵, 𝐶, 𝐷]. Proof. For 𝛼 = 0, the result is obvious. So we assume that 𝛼 > 0 and using the integral representation for 𝑓(𝑧) ∈ 𝛽 − UQ𝛼 [𝐴, 𝐵, 𝐶, 𝐷], we have 𝑓 (𝑧) =
1 1−1/𝛼 𝑧 2−1/𝛼 𝐹 (𝑡) 𝑑𝑡, 𝑧 ∫ 𝑡 𝛼 0
(𝑧 ∈ U) ,
(56)
where 𝐹(𝑧) ∈ 𝛽 − UK[𝐴, 𝐵, 𝐶, 𝐷], since 𝑓(𝑧) = 𝜙𝛼 (𝑧) ∗ 𝐹(𝑧), where ∞
1 𝑧𝑛 , 𝛼 − 1) + 1 (𝑛 𝑛=1
𝜙𝛼 (𝑧) = ∑
(𝑧 ∈ U)
(57)
is a convex function in U. Now using Lemma 8, we conclude that 𝑓(𝑧) ∈ 𝛽 − UK[𝐴, 𝐵, 𝐶, 𝐷]. This completes the proof. Theorem 22. If 0 ≤ 𝛼1 ≤ 𝛼2 , Then, 𝛽 − UQ𝛼2 [𝐴, 𝐵, 𝐶, 𝐷] ⊂ 𝛽 − UQ𝛼1 [𝐴, 𝐵, 𝐶, 𝐷] .
(58)
Proof. Let 𝑓(𝑧) ∈ 𝛽 − UQ𝛼2 [𝐴, 𝐵, 𝐶.𝐷]. Then
(𝑧𝑓 (𝑧)) ] ∈ 𝛽 − P [𝐴, 𝐵] . [(1 − 𝛼2 ) 𝑓 (𝑧) + 𝛼2 𝑔 (𝑧) 𝑔 (𝑧) ] [ Now consider
(59)
(𝑧𝑓 (𝑧)) 𝑓 (𝑧) ] + 𝛼1 𝑗 (𝛼1 , 𝑓; 𝑧) = [(1 − 𝛼1 ) 𝑔 (𝑧) 𝑔 (𝑧) ] [ = (1 −
+
𝛼1 𝑓 (𝑧) ) 𝛼2 𝑔 (𝑧)
𝑓 (𝑧) 𝛼1 [ + 𝛼2 (1 − 𝛼2 ) 𝛼2 𝑔 (𝑧) [
= (1 −
(𝑧𝑓 (𝑧)) 𝑔 (𝑧)
(60)
] ]
𝛼1 𝛼 ) 𝑗 (0, 𝑓; 𝑧) + 1 𝑗 (𝛼2 , 𝑓; 𝑧) . 𝛼2 𝛼2
Since 𝛽 − P[𝐴, 𝐵] is convex, (1 −
𝛼 𝛼1 ) 𝑗 (0, 𝑓; 𝑧) + 1 𝑗 (𝛼2 , 𝑓; 𝑧) ∈ 𝛽 − P [𝐴, 𝐵] 𝛼2 𝛼2
and consequently 𝑓(𝑧) completes the proof.
∈
(61)
𝛽 − UQ𝛼1 [𝐴, 𝐵, 𝐶, 𝐷]. This
Conflicts of Interest The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments The authors would like to acknowledge Professor Dr. Salim ur Rehman, V.C. Sarhad University of Science & IT, for providing excellent research and academic environment.
References [1] A. W. Goodman, Univalent Functions, vol. 1-2, Polygonal Publishing House, Washington, DC, USA, 1983. [2] W. Janowski, “Some extremal problems for certain families of analytic functions,” Annales Polonici Mathematici, vol. 28, pp. 297–326, 1973. [3] S. Kanas and W. Wi´sniowska, “Conic regions and k-uniform convexity,” Journal of Computational and Applied Mathematics, vol. 105, pp. 327–336, 1999. [4] S. Kanas and W. Wi´sniowska, “Conic domains and starlike functions,” Revue Roumaine de Math´ematiques Pures et Appliqu´ees, vol. 45, pp. 647–657, 2000. [5] K. I. Noor and S. N. Malik, “On coefficient inequalities of functions associated with conic domains,” Computers & Mathematics with Applications, vol. 62, pp. 2209–2217, 2011. [6] S. Mahmood, M. Arif, and S. N. Malik, “Janowski type closeto-convex functions associated with conic regions,” Journal of Inequalities and Applications, vol. 2017, article 259, 2017. [7] K. I. Noor, M. Arif, and W. Ul-Haq, “On k-uniformly close-toconvex functions of complex order,” Applied Mathematics and Computation, vol. 215, no. 2, pp. 629–635, 2009. [8] K. I. Noor and A. Aboudi, “Alpha quasi convex univalent functions,” Caribbean Journal of Mathematics, vol. 3, pp. 1–8, 1984. [9] C. Selvaraj and G. Thirupathi, “Certain generalization of Noor functions,” International Journal of Mathematical Analysis, vol. 9, no. 17, pp. 847–855, 2015. [10] S. S. Miller and P. T. Mocanu, Differential Subordinations: Theory and Applications, Pure and Applied Mathematics no. 225, Dekker, New York, NY, USA, 2000.
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