Neighborhoods of Certain Multivalently Analytic Functions

Abstract

We introduce and investigate two new general subclasses of multivalently analytic functions of complex order by making use of the familiar convolution structure of analytic functions. Among the various results obtained here for each of these function classes, we derive the coefficient bounds, distortion inequalities, and other interesting properties and characteristics for functions belonging to the classes introduced here.

1. Introduction and Definitions

Let ℝ=(−∞,∞) be the set of real numbers, and let ℂ be the set of complex numbers, ℕℕ∶={1,2,3,…}=ℕ₀⧵{0},_∗∶=ℕ⧵{1}={2,3,4,…}.(1.1)

Let 𝒯_𝑝 denote the class of functions of the form 𝑓(𝑧)=𝑧^𝑝−^∞_𝑗=𝑘𝑎_𝑗𝑧^𝑗𝑝<𝑘;𝑎_𝑗≥0(𝑗≥𝑘);𝑘,𝑝∈ℕ,(1.2) which are analytic and 𝑝-valent in the open unit disk 𝕌={𝑧∶𝑧∈ℂ,|𝑧|<1}.(1.3)

Denote by 𝑓∗𝑔 the Hadamard product (or convolution) of the functions 𝑓 and 𝑔, that is, if 𝑓 is given by (1.2) and 𝑔 is given by 𝑔(𝑧)=𝑧^𝑝+^∞_𝑗=𝑘𝑏_𝑗𝑧^𝑗𝑝<𝑘;𝑏_𝑗≥0(𝑗≥𝑘);𝑘,𝑝∈ℕ,(1.4) then (𝑓∗𝑔)(𝑧)∶=𝑧^𝑝−^∞_𝑗=𝑘𝑎_𝑗𝑏_𝑗𝑧^𝑗=∶(𝑔∗𝑓)(𝑧).(1.5)

In [1], the author defined the following general class.

Definition 1.1. Let the function 𝑓∈𝒯_𝑝. Then we say that 𝑓 is in the class 𝒮_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛) if it satisfies the condition |||||1𝑏𝑧^𝑛ℱ_𝜆,𝜇∗𝑔^(𝑚+𝑛)(𝑧)𝑚+𝑛<𝑝<𝑘;𝑝,𝑛∈ℕ;𝑚ℱ∈ℕ_𝜆,𝜇0∗𝑔^(𝑚)(𝑧)−(𝑝−𝑚)^𝑛|||||<𝛽,;𝑏∈ℂ⧵{0};0≤𝜇≤𝜆≤1;0<𝛽≤1;𝑧∈𝕌(1.6) where ℱ_𝜆,𝜇(𝑧)=𝜆𝜇𝑧²𝑓^(𝑧)+(𝜆−𝜇)𝑧𝑓^(𝑧)+(1−𝜆+𝜇)𝑓(𝑧),(1.7)𝑔 is given by (1.4), and (𝜈)_𝑛 denotes the falling factorial defined as follows: (𝜈)_𝑛(𝜈)₀=1=∶𝜈0,=𝜈(𝜈−1)⋯(𝜈−𝑛+1)=∶𝑛!𝜈𝑛(𝑛∈ℕ).(1.8)

Various special cases of the class 𝒮_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛) were considered by many earlier researchers on this topic of geometric function theory. For example, 𝒮_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛) reduces to the function class(i)𝒮^𝑛𝑔(𝑝,𝜆,𝑏,𝛽) for 𝑚=0, 𝑛=1, and 𝜇=0, studied by Mostafa and Aouf [2],(ii)𝒮_𝑔(𝑝,𝑘,𝑏,𝑚,𝑛) for 𝜆=𝜇=0, and 𝛽=1, studied by Srivastava et al. [3],(iii)𝒮_𝑔(𝑝,𝑛,𝑏,𝑚) for 𝑛=1, 𝜆=𝜇=0, and 𝛽=1, studied by Prajapat et al. [4],(iv)𝒮_𝑛,𝑝(𝑔;𝜆,𝜇,𝛼) for 𝑚=0, 𝑛=1, 𝛽=1, and 𝑏=𝑝(1−𝛼)(0≤𝛼<1), studied by Srivastava and Bulut [5],(v)𝒯𝒮^∗𝑔(𝑝,𝑚,𝛼) for 𝑚=0, 𝑛=1, 𝜆=𝜇=0, 𝛽=1, and 𝑏=𝑝(1−𝛼)(0≤𝛼<1), studied by Ali et al. [6].

Definition 1.2. Let the function 𝑓∈𝒯_𝑝. Then we say that 𝑓 is in the class 𝒦_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛;𝑞,𝑢) if it satisfies the following nonhomogenous Cauchy-Euler differential equation (see, e.g., [7, page 1360, Equation (9)] and [5, page 6512, Equation (1.9)]): 𝑧^𝑞𝑑^𝑞𝑤𝑑𝑧^𝑞+1𝑞(𝑢+𝑞−1)𝑧^𝑞−1𝑑^𝑞−1𝑤𝑑𝑧^𝑞−1+⋯+𝑞𝑞𝑤^𝑞−1_𝜀=0(𝑢+𝜀)=ℎ(𝑧)^𝑞−1_𝜀=0(𝑢+𝜀+𝑝),(1.9) where 𝑤=𝑓(𝑧)∈𝒯_𝑝;ℎ∈𝒮_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛);𝑞∈ℕ^∗,𝑢∈(−𝑝,∞).(1.10)

Setting 𝑚=0, 𝑛=1, 𝜇=0, and 𝑞=2 in Definition 1.2, we have the special class introduced by Mostafa and Aouf [2].

Following the works of Goodman [8] and Ruscheweyh [9] (see also [10, 11]), Altıntaş [12] defined the 𝛿-neighborhood of a function 𝑓∈𝒯(𝑝) by 𝒩^𝛿𝑘(𝑓)=ℎ∈𝒯_𝑝∶ℎ(𝑧)=𝑧^𝑝−^∞_𝑗=𝑘𝑐_𝑗𝑧^𝑗,^∞_𝑗=𝑘𝑗||𝑎_𝑗−𝑐_𝑗||≤𝛿.(1.11)

It follows from the definition (1.11) that if 𝑒(𝑧)=𝑧^𝑝(𝑝∈ℕ),(1.12) then 𝒩^𝛿𝑘(𝑒)=ℎ∈𝒯_𝑝∶ℎ(𝑧)=𝑧^𝑝−^∞_𝑗=𝑘𝑐_𝑗𝑧^𝑗,^∞_𝑗=𝑘𝑗||𝑐_𝑗||≤𝛿.(1.13)

The main object of this paper is to investigate the various properties and characteristics of functions belonging to the above-defined classes 𝒮_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛),𝒦_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛;𝑞,𝑢).(1.14) Apart from deriving coefficient bounds and distortion inequalities for each of these classes, we establish several inclusion relationships involving the 𝛿-neighborhoods of functions belonging to the general classes which are introduced above.

2. Coefficient Bounds and Distortion Theorems

Lemma 2.1 (see [1]). Let the function 𝑓∈𝒯_𝑝 be given by (1.2). Then 𝑓 is in the class 𝒮_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛) if and only if ^∞_𝑗=𝑘𝑚+(𝑛𝑗)<_𝑚𝑝(𝑗<−𝑘𝑚;𝑝),_𝑛𝑛−∈(𝑝ℕ−;𝑚𝑚)∈_𝑛+ℕ₀𝛽||𝑏||𝜓(𝑗)𝑎_𝑗𝑏_𝑗≤𝛽||𝑏||(𝑝)_𝑚𝜓(𝑝),;𝑏∈ℂ⧵{0};0<𝛽≤1;𝑧∈𝕌,(2.1) where 𝜓(𝑠)=(𝑠−1)(𝜆𝜇𝑠+𝜆−𝜇)+1(0≤𝜇≤𝜆≤1).(2.2)

Remark 2.2. If we set 𝑚=0, 𝑛=1, and 𝜇=0 in Lemma 2.1, then we have [2, Lemma 1].

Lemma 2.3 (See[1]). Let the function 𝑓∈𝒯_𝑝 given by (1.2) be in the class 𝒮_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛). Then, for 𝑏_𝑗≥𝑏_𝑘(𝑗≥𝑘), one has ^∞_𝑗=𝑘𝑎_𝑗≤𝛽||𝑏||(𝑝)^𝑚𝜓(𝑝)(𝑘)_𝑚(𝑘−𝑚)_𝑛−(𝑝−𝑚)_𝑛+𝛽||𝑏||𝜓(𝑘)𝑏_𝑘,(2.3)_∞_𝑗=𝑘𝑗𝑎_𝑗≤(𝑘−𝑚)!𝛽||𝑏||(𝑝)^𝑚𝜓(𝑝)(𝑘−1)!(𝑘−𝑚)_𝑛−(𝑝−𝑚)_𝑛+𝛽||𝑏||𝜓(𝑘)𝑏_𝑘𝑝>||𝑏||,(2.4) where 𝜓 is defined by (2.2).

Remark 2.4. If we set 𝑚=0, 𝑛=1, and 𝜇=0 in Lemma 2.3, then we have [2, Lemma  2].

The distortion inequalities for functions in the class 𝒮_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛) are given by the following Theorem 2.5.

Theorem 2.5. Let a function 𝑓∈𝒯_𝑝 be in the class 𝒮_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛). Then ||𝑓(𝑧)||≤|𝑧|^𝑝+𝛽||𝑏||(𝑝)^𝑚𝜓(𝑝)(𝑘)_𝑚(𝑘−𝑚)_𝑛−(𝑝−𝑚)_𝑛+𝛽||𝑏||𝜓(𝑘)𝑏_𝑘||𝑓(𝑧)||≥|𝑧|_𝑝|𝑧|^𝑘,(2.5)−𝛽||𝑏||(𝑝)^𝑚𝜓(𝑝)(𝑘)_𝑚(𝑘−𝑚)_𝑛−(𝑝−𝑚)_𝑛+𝛽||𝑏||𝜓(𝑘)𝑏_𝑘|𝑧|^𝑘,(2.6) and in general ||𝑓^(𝑟)(𝑧)||≤(𝑝)_𝑟|𝑧|^𝑝−𝑟+𝛽||𝑏||(𝑝)^𝑚(𝑘)_𝑟𝜓(𝑝)(𝑘)_𝑚(𝑘−𝑚)_𝑛−(𝑝−𝑚)_𝑛+𝛽||𝑏||𝜓(𝑘)𝑏_𝑘||𝑓_(𝑟)|𝑧|^𝑘−𝑟,(𝑧)||≥(𝑝)_𝑟|𝑧|^𝑝−𝑟−𝛽||𝑏||(𝑝)^𝑚(𝑘)_𝑟𝜓(𝑝)(𝑘)_𝑚(𝑘−𝑚)_𝑛−(𝑝−𝑚)_𝑛+𝛽||𝑏||𝜓(𝑘)𝑏_𝑘𝑝>𝑟;𝑟∈ℕ₀|𝑧|^𝑘−𝑟,;𝑧∈𝕌,(2.7) where 𝜓 is defined by (2.2).

Proof. Suppose that 𝑓∈𝒮_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛). We find from the inequality (2.3) that ||𝑓(𝑧)||≤|𝑧|^𝑝+|𝑧|^𝑘∞_𝑗=𝑘𝑎_𝑗≤|𝑧|^𝑝+𝛽||𝑏||(𝑝)^𝑚𝜓(𝑝)(𝑘)_𝑚(𝑘−𝑚)_𝑛−(𝑝−𝑚)_𝑛+𝛽||𝑏||𝜓(𝑘)𝑏_𝑘|𝑧|^𝑘,(2.8) which is equivalent to (2.5) and ||𝑓(𝑧)||≥|𝑧|^𝑝−|𝑧|^𝑘∞_𝑗=𝑘𝑎_𝑗≥|𝑧|^𝑝−𝛽||𝑏||(𝑝)^𝑚𝜓(𝑝)(𝑘)_𝑚(𝑘−𝑚)_𝑛−(𝑝−𝑚)_𝑛+𝛽||𝑏||𝜓(𝑘)𝑏_𝑘|𝑧|^𝑘,(2.9) which is precisely the assertion (2.6).

If we set 𝑚=0, 𝑛=1, and 𝜇=0 in Theorem 2.5, then we get the following.

Corollary 2.6. Let a function 𝑓∈𝒯_𝑝 be in the class 𝒮^𝑛𝑔(𝑝,𝜆,𝑏,𝛽). Then ||𝑓(𝑧)||≤|𝑧|^𝑝+𝛽||𝑏||[1+𝜆(𝑝−1)]𝑘−𝑝+𝛽||𝑏||[1+𝜆(𝑘−1)]𝑏_𝑘||𝑓(𝑧)||≥|𝑧|_𝑝|𝑧|^𝑘,−𝛽||𝑏||[1+𝜆(𝑝−1)]𝑘−𝑝+𝛽||𝑏||[1+𝜆(𝑘−1)]𝑏_𝑘|𝑧|^𝑘,(2.10) and in general ||𝑓^(𝑟)(𝑧)||≤𝑝!(𝑝−𝑟)!|𝑧|^𝑝−𝑟+𝑘!𝛽||𝑏||[1+𝜆(𝑝−1)](𝑘−𝑟)!𝑘−𝑝+𝛽||𝑏||[1+𝜆(𝑘−1)]𝑏_𝑘||𝑓_(𝑟)|𝑧|^𝑘−𝑟,(𝑧)||≥𝑝!(𝑝−𝑟)!|𝑧|^𝑝−𝑟−𝑘!𝛽||𝑏||[1+𝜆(𝑝−1)](𝑘−𝑟)!𝑘−𝑝+𝛽||𝑏||[1+𝜆(𝑘−1)]𝑏_𝑘𝑝>𝑟;𝑟∈ℕ₀|𝑧|^𝑘−𝑟,;𝑧∈𝕌,(2.11) where 𝜓 is defined by (2.2).

The distortion inequalities for functions in the class 𝒦_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛;𝑞,𝑢) are given by Theorem 2.7 below.

Theorem 2.7. Let a function 𝑓∈𝒯_𝑝 be in the class 𝒦_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛;𝑞,𝑢). Then ||𝑓(𝑧)||≤|𝑧|^𝑝+𝛽||𝑏||(𝑝)^𝑚𝜓(𝑝)∏^{𝑞−1𝜀=0}(𝑢+𝜀+𝑝)(𝑘)_𝑚(𝑘−𝑚)_𝑛−(𝑝−𝑚)_𝑛+𝛽||𝑏||𝜓(𝑘)(𝑞−1)∏^{𝑞−2𝜀=0}(𝑢+𝜀+𝑘)𝑏_𝑘||𝑓(𝑧)||≥|𝑧|_𝑝|𝑧|^𝑘,(2.12)−𝛽||𝑏||(𝑝)^𝑚𝜓(𝑝)∏^{𝑞−1𝜀=0}(𝑢+𝜀+𝑝)(𝑘)_𝑚(𝑘−𝑚)_𝑛−(𝑝−𝑚)_𝑛+𝛽||𝑏||𝜓(𝑘)(𝑞−1)∏^{𝑞−2𝜀=0}(𝑢+𝜀+𝑘)𝑏_𝑘|𝑧|^𝑘,(2.13) and in general ||𝑓^(𝑟)(𝑧)||≤(𝑝)_𝑟|𝑧|^𝑝−𝑟+(𝑘)^𝑟𝛽||𝑏||(𝑝)_𝑚𝜓(𝑝)∏^{𝑞−1𝜀=0}(𝑢+𝜀+𝑝)(𝑘)_𝑚(𝑘−𝑚)_𝑛−(𝑝−𝑚)_𝑛+𝛽||𝑏||𝜓(𝑘)(𝑞−1)∏^{𝑞−2𝜀=0}(𝑢+𝜀+𝑘)𝑏_𝑘||𝑓_(𝑟)|𝑧|^𝑘−𝑟,(𝑧)||≥(𝑝)_𝑟|𝑧|^𝑝−𝑟−(𝑘)^𝑟𝛽||𝑏||(𝑝)_𝑚𝜓(𝑝)∏^{𝑞−1𝜀=0}(𝑢+𝜀+𝑝)(𝑘)_𝑚(𝑘−𝑚)_𝑛−(𝑝−𝑚)_𝑛+𝛽||𝑏||𝜓(𝑘)(𝑞−1)∏^{𝑞−2𝜀=0}(𝑢+𝜀+𝑘)𝑏_𝑘𝑝>𝑟;𝑟∈ℕ₀|𝑧|^𝑘−𝑟,;𝑧∈𝕌,(2.14) where 𝜓 is defined by (2.2).

Proof. Suppose that a function 𝑓∈𝒯_𝑝 is given by (1.2), and also let the function ℎ∈𝒮_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛) be occurring in the nonhomogenous Cauchy-Euler differential equation (1.9) with of course 𝑐_𝑗≥0(𝑗≥𝑘).(2.15) Then we readily see from (1.9) that 𝑎_𝑗=∏^{𝑞−1𝜀=0}(𝑢+𝜀+𝑝)∏^{𝑞−1𝜀=0}(𝑢+𝜀+𝑗)𝑐^𝑗(𝑗≥𝑘),(2.16) so that 𝑓(𝑧)=𝑧^𝑝−^∞_𝑗=𝑘𝑎_𝑗𝑧^𝑗=𝑧^𝑝−^∞_𝑗=𝑘∏^{𝑞−1𝜀=0}(𝑢+𝜀+𝑝)∏^{𝑞−1𝜀=0}(𝑢+𝜀+𝑗)𝑐^𝑗||𝑓(𝑧)||≤|𝑧|_𝑝𝑧^𝑗,(2.17)+|𝑧|^𝑘∞_𝑗=𝑘∏^{𝑞−1𝜀=0}(𝑢+𝜀+𝑝)∏^{𝑞−1𝜀=0}(𝑢+𝜀+𝑗)𝑐^𝑗.(2.18) Moreover, since ℎ∈𝒮_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛), the first assertion (2.3) of Lemma 2.3 yields the following inequality: 𝑐_𝑗≤𝛽||𝑏||(𝑝)^𝑚𝜓(𝑝)(𝑘)_𝑚(𝑘−𝑚)_𝑛−(𝑝−𝑚)_𝑛+𝛽||𝑏||𝜓(𝑘)𝑏_𝑘,(2.19) and together with (2.19) and (2.18) it yields that ||𝑓(𝑧)||≤|𝑧|^𝑝+𝛽||𝑏||(𝑝)^𝑚𝜓(𝑝)∏^{𝑞−1𝜀=0}(𝑢+𝜀+𝑝)(𝑘)_𝑚(𝑘−𝑚)_𝑛−(𝑝−𝑚)_𝑛+𝛽||𝑏||𝜓(𝑘)𝑏_𝑘|𝑧|^𝑘∞_𝑗=𝑘1∏^{𝑞−1𝜀=0}(𝑢+𝜀+𝑗).(2.20) Finally, in view of the following sum: ^∞_𝑗=𝑘1∏^{𝑞−1𝜀=0}(𝑗+𝑢+𝜀)=^∞_𝑗=𝑘^𝑞−1_𝜀=0(−1)^𝜀(𝑞−1−𝜀)!𝜀!(𝑗+(𝑢𝑢∈+ℝ𝜀)−{=−𝑘(,𝑞−−𝑘1−)∏1,−1𝑘^{𝑞−2𝜀=0}(−𝑢+2,𝜀…+}𝑘)),,(2.21) the assertion (2.12) of Theorem 2.7 follows at once from (2.20) together with (2.21). The assertion (2.13) can be proven by similarly applying (2.17), and (2.19)–(2.21).

Remark 2.8. If we set 𝑚=0, 𝑛=1, 𝜇=0, and 𝑞=2 in Theorem 2.7, then we have [2, Theorem 1].

3. Neighborhoods for the Classes 𝑆_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛) and 𝒦_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛;𝑞,𝑢)

In this section, we determine inclusion relations for the classes 𝒮_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛),𝒦_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛;𝑞,𝑢),(3.1) involving 𝛿-neighborhoods defined by (1.11) and (1.13).

Theorem 3.1 (see [1]). If 𝑏_𝑗≥𝑏_𝑘(𝑗≥𝑘) and 𝛿=(𝑘−𝑚)!𝛽||𝑏||(𝑝)^𝑚𝜓(𝑝)(𝑘−1)!(𝑘−𝑚)_𝑛−(𝑝−𝑚)_𝑛+𝛽||𝑏||𝜓(𝑘)𝑏_𝑘𝑝>||𝑏||,(3.2) then 𝒮_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛)⊂𝒩^𝛿𝑘(𝑒),(3.3) where 𝑒 and 𝜓 are given by (1.12) and (2.2), respectively.

Remark 3.2. If we set 𝑚=0, 𝑛=1, and 𝜇=0 in Theorem 3.1, then we have [2, Theorem 2].

Theorem 3.3. If 𝑏_𝑗≥𝑏_𝑘(𝑗≥𝑘) and 𝛿=(𝑘−𝑚)!𝛽||𝑏||(𝑝)^𝑚𝜓(𝑝)(𝑘−1)!(𝑘−𝑚)_𝑛−(𝑝−𝑚)_𝑛+𝛽||𝑏||𝜓(𝑘)𝑏_𝑘1+∏^{𝑞−1𝜀=0}(𝑢+𝜀+𝑝)(𝑞−1)∏^{𝑞−2𝜀=0}(𝑢+𝜀+𝑘)𝑝>||𝑏||,(3.4) then 𝒦_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛;𝑞,𝑢)⊂𝒩^𝛿𝑘(ℎ),(3.5) where ℎ and 𝜓 are given by (1.11) and (2.2), respectively.

Proof. Suppose that 𝑓∈𝒦_𝑔(𝑝,𝑘,𝜆,𝜇,𝑏,𝛽,𝑚,𝑛;𝑞,𝑢). Then, upon substituting from (2.16) into the following coefficient inequality: ^∞_𝑗=𝑘𝑗||𝑐_𝑗−𝑎_𝑗||≤^∞_𝑗=𝑘𝑗𝑐_𝑗+^∞_𝑗=𝑘𝑗𝑎_𝑗𝑐_𝑗≥0;𝑎_𝑗≥0,(3.6) we obtain ^∞_𝑗=𝑘𝑗||𝑐_𝑗−𝑎_𝑗||≤^∞_𝑗=𝑘𝑗𝑐_𝑗+^∞_𝑗=𝑘∏^{𝑞−1𝜀=0}(𝑢+𝜀+𝑝)∏^{𝑞−1𝜀=0}(𝑢+𝜀+𝑗)𝑗𝑐^𝑗.(3.7) Since ℎ∈𝒮_𝑔(𝑝,𝑘,𝜆,𝑏,𝛽,𝑚,𝑛), the assertion (2.4) of Lemma 2.3 yields 𝑗𝑐_𝑗≤(𝑘−𝑚)!𝛽||𝑏||(𝑝)^𝑚𝜓(𝑝)(𝑘−1)!(𝑘−𝑚)_𝑛−(𝑝−𝑚)_𝑛+𝛽||𝑏||𝜓(𝑘)𝑏_𝑘𝑝>||𝑏||.(3.8) Finally, by making use of (2.4) as well as (3.8) on the right-hand side of (3.7), we find that ^∞_𝑗=𝑘𝑗||𝑐_𝑗−𝑎_𝑗||≤(𝑘−𝑚)!𝛽||𝑏||(𝑝)^𝑚𝜓(𝑝)(𝑘−1)!(𝑘−𝑚)_𝑛−(𝑝−𝑚)_𝑛+𝛽||𝑏||𝜓(𝑘)𝑏_𝑘×1+^∞_𝑗=𝑘∏^{𝑞−1𝜀=0}(𝑢+𝜀+𝑝)∏^{𝑞−1𝜀=0}(𝑢+𝜀+𝑗),(3.9) which, by virtue of the sum in (2.21), immediately yields ^∞_𝑗=𝑘𝑗||𝑐_𝑗−𝑎_𝑗||≤(𝑘−𝑚)!𝛽||𝑏||(𝑝)^𝑚𝜓(𝑝)(𝑘−1)!(𝑘−𝑚)_𝑛−(𝑝−𝑚)_𝑛+𝛽||𝑏||𝜓(𝑘)𝑏_𝑘×1+∏^{𝑞−1𝜀=0}(𝑢+𝜀+𝑝)(𝑞−1)∏^{𝑞−2𝜀=0}(𝑢+𝜀+𝑘)=∶𝛿.(3.10) Thus, by applying the definition (1.11), we complete the proof of Theorem 3.3.

Remark 3.4. If we set 𝑚=0, 𝑛=1, 𝜇=0, and 𝑞=2 in Theorem 3.3, then we have [2, Theorem  3].

Acknowledgment

The present paper was supported by the Kocaeli University under Grant no. HD 2011/22.