On Certain Subclasses of Meromorphic Functions with Positive and Fixed Second Coefficients Involving the Liu-Srivastava Linear Operator

Abstract

We introduce and study a subclass Σ_𝑃(𝛾,𝑘,𝜆,𝑐) of meromorphic univalent functions defined by certain linear operator involving the generalized hypergeometric function. We obtain coefficient estimates, extreme points, growth and distortion inequalities, radii of meromorphic starlikeness, and convexity for the class Σ_𝑃(𝛾,𝑘,𝜆,𝑐) by fixing the second coefficient. Further, it is shown that the class Σ_𝑃(𝛾,𝑘,𝜆) is closed under convex linear combination.

1. Introduction

Let Σ denote the class of functions of the form 𝑓(𝑧)=𝑧⁻¹+^∞_𝑛=1𝑎_𝑛𝑧^𝑛,(1.1) which are analytic in the punctured unit disk𝕌^∗∶={𝑧∶𝑧∈ℂ,0<|𝑧|<1}=∶𝕌⧵{0}.(1.2)

Let Σ_𝒮,  Σ^∗(𝛾) and Σ_𝐾(𝛾),  (0≤𝛾<1) denote the subclasses of Σ that are meromorphically univalent, meromorphically starlike functions of order 𝛾, and meromorphically convex functions of order 𝛾, respectively. Analytically, 𝑓∈Σ^∗(𝛾) if and only if, 𝑓 is of the form (1.1) and satisfies−ℜ𝑧𝑓^(𝑧)𝑓(𝑧)>𝛾,𝑧∈𝕌,(1.3) similarly, 𝑓∈Σ_𝐾(𝛾), if and only if, 𝑓 is of the form (1.1) and satisfies −ℜ1+𝑧𝑓^(𝑧)𝑓^(𝑧)>𝛾,𝑧∈𝕌,(1.4) and similar other classes of meromorphically univalent functions have been extensively studied by Aouf and Darwish [1], Aouf and Joshi [2], Mogra et al. [3], Uralegaddi and Ganigi [4], Uralegaddi and Somanatha[5], and Owa and Pascu [6].

Let 𝑓, 𝑔∈Σ, where 𝑓(𝑧) is given by (1.1) and 𝑔(𝑧) is defined by𝑔(𝑧)=𝑧⁻¹+^∞_𝑛=1𝑏_𝑛𝑧^𝑛.(1.5)

Then the Hadamard product (or convolution) 𝑓∗𝑔 of the functions 𝑓(𝑧) and 𝑔(𝑧) is defined by(𝑓∗𝑔)(𝑧)∶=𝑧⁻¹+^∞_𝑛=1𝑎_𝑛𝑏_𝑛𝑧^𝑛=∶(𝑔∗𝑓)(𝑧).(1.6) Let Σ_𝑃 be the class of functions of the form 𝑓(𝑧)=𝑧⁻¹+^∞_𝑛=1𝑎_𝑛𝑧^𝑛,𝑎_𝑛≥0,(1.7) that are analytic and univalent in 𝕌^∗.

For complex parameters 𝛼₁,…,𝛼_𝑙 and 𝛽₁,…,𝛽_𝑚(𝛽_𝑗≠0,−1,…;𝑗=1,2,…,𝑚) the generalized hypergeometric function _𝑙𝐹_𝑚(𝑧) is defined by _𝑙𝐹_𝑚(𝑧)≡_𝑙𝐹_𝑚𝛼₁,…𝛼_𝑙;𝛽₁,…,𝛽_𝑚;𝑧∶=^∞_𝑛=0𝛼₁_𝑛…𝛼_𝑙_𝑛𝛽₁_𝑛…𝛽_𝑚_𝑛𝑧^𝑛𝑙≤𝑚+1;𝑙,𝑚∈ℕ𝑛₀!,∶=ℕ∪{0};𝑧∈𝑈,(1.8) where ℕ denotes the set of all positive integers and (𝜃)_𝑛 is the Pochhammer symbol defined by(𝜃)_𝑛=Γ(𝜃+𝑛)Γ(𝜃)=1𝜃,(𝜃+1)(𝜃+2)⋅(𝜃+𝑛−1),𝑛𝑛∈=ℕ0;;𝜃𝜃∈∈ℂℂ.⧵{0},(1.9)

Corresponding to a function _𝑙𝐹_𝑚(𝛼₁,…𝛼_𝑙;𝛽₁,…,𝛽_𝑚;𝑧) defined byℱ𝛼₁,…𝛼_𝑙;𝛽₁,…,𝛽_𝑚;𝑧∶=𝑧^𝑙−1𝐹_𝑚𝛼₁,…𝛼_𝑙;𝛽₁,…,𝛽_𝑚;𝑧.(1.10)

Liu and Srivastava [7] considered a linear operator ℋ(𝛼₁,…𝛼_𝑙;𝛽₁,…,𝛽_𝑚)∶Σ→Σ defined by the following Hadamard product (or convolution):ℋ𝛼₁,…𝛼_𝑙;𝛽₁,…,𝛽_𝑚𝑓(𝑧)==𝑧ℱ₋₁𝛼₁,…𝛼_𝑙;𝛽₁,…,𝛽_𝑚;𝑧∗𝑓(𝑧)+^∞_𝑛=1|||||𝛼₁_𝑛+1…𝛼_𝑙_𝑛+1𝛽₁_𝑛+1…𝛽_𝑚_𝑛+1|||||𝑎_𝑛𝑧^𝑛(𝑛+1)!,(1.11) where, 𝛼_𝑖>0, (𝑖=1,2,…𝑙), 𝛽_𝑗>0, (𝑗=1,2,…𝑚), 𝑙≤𝑚+1;  𝑙, 𝑚∈ℕ₀=ℕ∪{0}. For notational simplicity, we use a shorter notations ℋ^𝑙𝑚[𝛼₁] for ℋ(𝛼₁,…𝛼_𝑙;𝛽₁,…,𝛽_𝑚) and Γ_𝑛𝛼₁=𝛼¹_𝑛+1…𝛼_𝑙_𝑛+1𝛽₁_𝑛+1…𝛽_𝑚_𝑛+11(𝑛+1)!,(1.12) unless otherwise stated in the sequel. We note that the linear operator 𝐻^𝑙𝑚[𝛼₁] was earlier defined for multivalent functions by Dziok and Srivastava [8] and was investigated by Liu and Srivastava [7].

Making use of the operator ℋ^𝑙𝑚[𝛼₁], now we consider a subclass of functions in Σ_𝑃 as follows.

Definition 1.1. For 0≤𝛾<1 and 𝑘≥0 and 0≤𝜆<1/2, let Σ(𝛾,𝑘,𝜆) denote a subclass of Σ consisting functions of the form (1.1) satisfying the condition that −ℜ𝑧ℋ𝛼¹𝑓(𝑧)′+𝜆𝑧²ℋ𝛼₁𝑓(𝑧)^>𝑘|||||(𝑧1ℋ−𝜆)𝛼ℋ₁𝛼₁𝑓(𝑧)+𝜆𝑧ℋ𝛼₁𝑓(𝑧)′+𝛾𝑓(𝑧)′+𝜆𝑧²ℋ𝛼₁𝑓(𝑧)^(1−𝜆)ℋ𝛼₁𝑓(𝑧)+𝜆𝑧ℋ𝛼₁𝑓(𝑧)′+1|||||,𝑧∈𝕌^∗,(1.13) where ℋ[𝛼₁]𝑓(𝑧) is given by (1.11). Furthermore, we say that a function 𝑓∈Σ_𝑃(𝛾,𝑘,𝜆), whenever 𝑓(𝑧) is of the form (1.7).
We observe that, by specializing the parameters 𝑙,  𝑚,  𝛼₁,…,𝛼_𝑙,  𝛽₁,…,𝛽_𝑚,  𝑘,  𝛾 and 𝜆 the class leads to various subclasses. As for illustrations, we present some examples for the cases.

Example 1.2. If 𝑙=2 and 𝑚=1 with 𝛼₁=1, 𝛼₂=1, 𝛽₁=1 and  𝑓(𝑧)  of the form (1.7), then we obtain the new subclass ℳ_𝑃(𝛾,𝑘,𝜆)   defined by −ℜ𝑧𝑓^(𝑧)+𝜆𝑧²𝑓^(𝑧)(1−𝜆)𝑓(𝑧)+𝜆𝑧𝑓^(𝑧)+𝛾>𝑘||||𝑧𝑓^(𝑧)+𝜆𝑧²𝑓^(𝑧)(1−𝜆)𝑓(𝑧)+𝜆𝑧𝑓^(𝑧)+1||||,𝑧∈𝕌^∗.(1.14)

Example 1.3. For 𝑙=2, 𝑚=1, 𝛼₁=𝛿+1, 𝛽₁=𝛼₂=1, and𝑓(𝑧) of the form (1.7), then we get the new subclass 𝒟^𝛿𝑃(𝛾,𝑘,𝜆) defined by −ℜ⎧⎪⎨⎪⎩𝑧𝒟^𝛿𝑓(𝑧)^+𝜆𝑧²𝒟^𝛿𝑓(𝑧)^(1−𝜆)𝒟^𝛿𝑓(𝑧)+𝜆𝑧𝒟^𝛿𝑓(𝑧)′+𝛾⎫⎪⎬⎪⎭>𝑘|||||𝑧𝒟^𝛿𝑓(𝑧)^+𝜆𝑧²𝒟^𝛿𝑓(𝑧)^(1−𝜆)𝒟^𝛿𝑓(𝑧)+𝜆𝑧𝒟^𝛿𝑓(𝑧)′+1|||||,𝑧∈𝕌^∗,(1.15) where 𝒟^𝛿𝑓(𝑧)=1/(𝑧(1−𝑧)^𝛿+1)∗𝑓(𝑧)(𝛿>−1), is the differential operator which was introduced by Ganigi and Uralegaddi [9]. Also we note that the class 𝒟^𝛿𝑃(𝛾,𝑘,𝜆) was introduced by Atshan and Kulkarni [10].

Example 1.4. For 𝑙=2, 𝑚=1,  𝛼₂=1, and 𝑓(𝑧) of the form (1.7), then we obtain the new subclass ℒ_𝑃(𝛾,𝑘,𝜆) defined by −ℜ𝑧ℒ𝛼¹,𝛽₁𝑓(𝑧)′+𝜆𝑧²ℒ𝛼₁,𝛽₁𝑓(𝑧)^>𝑘|||||(𝑧1ℒ−𝜆𝛼)ℒ₁𝛼₁,𝛽₁𝑓(𝑧)+𝜆𝑧ℒ𝛼₁,𝛽₁𝑓(𝑧)′+𝛾,𝛽₁𝑓(𝑧)′+𝜆𝑧²ℒ𝛼₁,𝛽₁𝑓(𝑧)^(1−𝜆)ℒ𝛼₁,𝛽₁𝑓(𝑧)+𝜆𝑧ℒ𝛼₁,𝛽₁𝑓(𝑧)′+1|||||,𝑧∈𝕌^∗,(1.16) where the operator ℒ[𝛼₁;𝛽₁]𝑓(𝑧) was introduced and studied by Liu and Srivastava [11] (see also [12, 13]).
For the class Σ_𝑃(𝛾,𝑘,𝜆), the following characterization was given by Magesh et al. [14].

Theorem 1.5. Let 𝑓∈Σ_𝑃 be given by (1.7). Then 𝑓∈Σ_𝑃(𝛾,𝑘,𝜆), if and only if, ^∞_𝑛=1(1+(𝑛−1)𝜆)[𝑛(𝑘+1)+(𝑘+𝛾)]Γ_𝑛𝛼₁𝑎_𝑛≤(1−𝛾)(1−2𝜆),(1.17) where Γ_𝑛(𝛼₁) is given by (1.12).

For a function defined by (1.7) and in the class Σ_𝑃(𝛾,𝑘,𝜆), Theorem 1.5, immediately yieldsΓ₁𝛼₁𝑎₁≤(1−𝛾)(1−2𝜆)(2𝑘+𝛾+1).(1.18) Hence we may takeΓ₁𝛼₁𝑎₁=(1−𝛾)(1−2𝜆)𝑐(2𝑘+𝛾+1),0<𝑐<1.(1.19)

Motivated by Aouf and Darwish [1], Aouf and Joshi [2], Ghanim and Darus [12], and Uralegaddi [15], we now introduce the following class of functions and use the similar techniques to prove our results.

Let Σ_𝑃(𝛾,𝑘,𝜆,𝑐) be the subclass of Σ_𝑃(𝛾,𝑘,𝜆) consisting functions in of the form𝑓(𝑧)=1𝑧+(1(−2𝑘𝛾)+(1𝛾−+21𝜆))𝑐𝑧+^∞_𝑛=2Γ_𝑛𝛼₁𝑎_𝑛𝑧^𝑛,0<𝑐<1.(1.20)

In this paper, coefficient estimates, extreme points, growth and distortion bounds, radii of meromorphically starlikeness and convexity are discussed for the class Σ_𝑃(𝛾,𝑘,𝜆) by fixing the second coefficient. Further, it is shown that the class Σ_𝑃(𝛾,𝑘,𝜆) is closed under convex linear combination.

2. Main Results

In our first theorem, we now find out the coefficient inequality for the class Σ_𝑃(𝛾,𝑘,𝜆,𝑐).

Theorem 2.1. Let the function 𝑓(𝑧) defined by (1.20). Then 𝑓∈Σ_𝑃(𝛾,𝑘,𝜆,𝑐), if and only if, ^∞_𝑛=2(1+(𝑛−1)𝜆)[𝑛(𝑘+1)+(𝑘+𝛾)]Γ_𝑛𝛼₁𝑎_𝑛≤(1−𝛾)(1−2𝜆)(1−𝑐).(2.1) The result is sharp.

Proof. By putting Γ₁𝛼₁𝑎₁=(1−𝛾)(1−2𝜆)𝑐(2𝑘+𝛾+1),0<𝑐<1,(2.2) in (1.17), the result is easily derived. The result is sharp for the function 𝑓(𝑧)=1𝑧+(1(−2𝑘𝛾)+(1𝛾−+21𝜆))𝑐𝑧+(1+(𝑛−1(1)𝜆−)[𝛾𝑛)((𝑘1−+21𝜆))+(1(𝑘−+𝑐)𝛾)]Γ_𝑛𝛼₁𝑧^𝑛,𝑛≥2.(2.3)

Corollary 2.2. If the function 𝑓 defined by (1.20) is in the class Σ_𝑃(𝛾,𝑘,𝜆,𝑐), then 𝑎_𝑛≤(1−𝛾)(1−2𝜆)(1−𝑐)(1+(𝑛−1)𝜆)[𝑛(𝑘+1)+(𝑘+𝛾)]Γ_𝑛𝛼₁,𝑛≥2.(2.4) The result is sharp for the function 𝑓(𝑧) given by (2.3).

Next we prove the following growth and distortion properties for the class Σ_𝑃(𝛾,𝑘,𝜆,𝑐).

Theorem 2.3. If the function 𝑓(𝑧) defined by (1.20) is in the class Σ_𝑃(𝛾,𝑘,𝜆,𝑐) for 0<|𝑧|=𝑟<1, then one has 1𝑟−(1(−2𝑘𝛾)+(1𝛾−+21𝜆))𝑐𝑟−(1(1−+𝛾)𝜆()1(3−𝑘2+𝜆)𝛾(1+−2)𝑐)𝑟²≤1≤||𝑓(𝑧)||𝑟+(1(−2𝑘𝛾)+(1𝛾−+21𝜆))𝑐𝑟+(1(1−+𝛾)𝜆()1(3−𝑘2+𝜆)𝛾(1+−2)𝑐)𝑟².(2.5) The result is sharp for the function 𝑓(𝑧) given by 𝑓(𝑧)=1𝑧+(1(−2𝑘𝛾)+(1𝛾−+21𝜆))𝑐𝑧+(1(1−+𝛾)𝜆()1(3−𝑘2+𝜆)𝛾(1+−2)𝑐)𝑧².(2.6)

Proof. Since Σ_𝑃(𝛾,𝑘,𝜆,𝑐), Theorem 2.1 yields Γ_𝑛𝛼₁𝑎_𝑛≤(1−𝛾)(1−2𝜆)(1−𝑐)(1+(𝑛−1)𝜆)[𝑛(𝑘+1)+(𝑘+𝛾)],𝑛≥2.(2.7) Thus, for 0<|𝑧|=𝑟<1, ||𝑓(𝑧)||≤1|𝑧|+(1(−2𝑘𝛾)+(1𝛾−+21𝜆))𝑐|𝑧|+^∞_𝑛=2Γ_𝑛𝛼₁𝑎_𝑛|𝑧|^𝑛≤1𝑟+(1(−2𝑘𝛾)+(1𝛾−+21𝜆))𝑐𝑟+𝑟^2∞_𝑛=2Γ_𝑛𝛼₁𝑎_𝑛≤1𝑟+(1(−2𝑘𝛾)+(1𝛾−+21𝜆))𝑐𝑟+(1(1−+𝛾)𝜆()1(3−𝑘2+𝜆)𝛾(1+−2)𝑐)𝑟²||𝑓(𝑧)||≥1,|𝑧|−(1(−2𝑘𝛾)+(1𝛾−+21𝜆))𝑐|𝑧|−^∞_𝑛=2Γ_𝑛𝛼₁𝑎_𝑛|𝑧|^𝑛≥1𝑟−(1(−2𝑘𝛾)+(1𝛾−+21𝜆))𝑐𝑟−𝑟^2∞_𝑛=2Γ_𝑛𝛼₁𝑎_𝑛≥1𝑟−(1(−2𝑘𝛾)+(1𝛾−+21𝜆))𝑐𝑟−(1(1−+𝛾)𝜆()1(3−𝑘2+𝜆)𝛾(1+−2)𝑐)𝑟².(2.8) Thus the proof of the theorem is complete.

Theorem 2.4. If the function 𝑓(𝑧) defined by (1.20) is in the class Σ_𝑃(𝛾,𝑘,𝜆,𝑐) for 0<|𝑧|=𝑟<1, then one has 1𝑟²−(1−𝛾)(1−2𝜆)𝑐(≤2𝑘||+𝑓𝛾+1)−(1(1−+𝛾)𝜆()1(3−𝑘2+𝜆)𝛾(1+−2)_𝑐)𝑟(𝑧)||≤1𝑟²+(1−𝛾)(1−2𝜆)𝑐(2𝑘+𝛾+1)+(1(1−+𝛾)𝜆()1(3−𝑘2+𝜆)𝛾(1+−2)𝑐)𝑟.(2.9) The result is sharp for the function 𝑓(𝑧) given by 𝑓(𝑧)=1𝑧+(1(−2𝑘𝛾)+(1𝛾−+21𝜆))𝑐𝑧+(1(1−+𝛾)𝜆()1(3−𝑘2+𝜆)𝛾(1+−2)𝑐)𝑧².(2.10)

Proof. In view of Theorem 2.1, it follows that 𝑛Γ_𝑛𝛼₁𝑎_𝑛≤𝑛(1−𝛾)(1−2𝜆)(1−𝑐)(1+(𝑛−1)𝜆)[𝑛(𝑘+1)+(𝑘+𝛾)],𝑛≥2.(2.11) Thus, for 0<|𝑧|=𝑟<1 and making use of (2.11), we obtain ||𝑓^(𝑧)||≤|||−1𝑧²|||+(1−𝛾)(1−2𝜆)𝑐(2𝑘+𝛾+1)+^∞_𝑛=2𝑛Γ_𝑛𝛼₁𝑎_𝑛|𝑧|^𝑛−1≤1𝑟²+(1−𝛾)(1−2𝜆)𝑐(2𝑘+𝛾+1)+𝑟^∞_𝑛=2𝑛Γ_𝑛𝛼₁𝑎_𝑛≤1𝑟²+(1−𝛾)(1−2𝜆)𝑐||𝑓(2𝑘+𝛾+1)+(1(1−+𝛾)𝜆()1(3−𝑘2+𝜆)𝛾(1+−2)𝑐)𝑟,_(𝑧)||≥|||−1𝑧²|||−(1−𝛾)(1−2𝜆)𝑐(2𝑘+𝛾+1)−^∞_𝑛=2𝑛Γ_𝑛𝛼₁𝑎_𝑛|𝑧|^𝑛−1≥1𝑟²−(1−𝛾)(1−2𝜆)𝑐(2𝑘+𝛾+1)−𝑟^∞_𝑛=2𝑛Γ_𝑛𝛼₁𝑎_𝑛≥1𝑟²−(1−𝛾)(1−2𝜆)𝑐(2𝑘+𝛾+1)−(1(1−+𝛾)𝜆()1(3−𝑘2+𝜆)𝛾(1+−2)𝑐)𝑟.(2.12)

Hence the result follows.

Next, we will show that the class Σ_𝑃(𝛾,𝑘,𝜆,𝑐) is closed under convex linear combination.

Theorem 2.5. If 𝑓₁(𝑧)=1𝑧+(1(−2𝑘𝛾)+(1𝛾−+21𝜆))𝑐𝑧,(2.13) and for 𝑛≥2𝑓_𝑛(𝑧)=1𝑧+(1(−2𝑘𝛾)+(1𝛾−+21𝜆))𝑐𝑧+^∞_𝑛=2(1−𝛾)(1−2𝜆)(1−𝑐)(1+(𝑛−1)𝜆)[𝑛(𝑘+1)+(𝑘+𝛾)]Γ_𝑛𝛼₁𝑧^𝑛.(2.14) Then 𝑓∈Σ_𝑃(𝛾,𝑘,𝜆,𝑐) if and only if it can expressed in the form 𝑓(𝑧)=^∞_𝑛=1𝜇_𝑛𝑓_𝑛(𝑧),(2.15) where 𝜇_𝑛≥0 and ∑^∞𝑛=1𝜇_𝑛≤1.

Proof. From (2.13), (2.14), and (2.15), we have 𝑓(𝑧)=1𝑧+(1(−2𝑘𝛾)+(1𝛾−+21𝜆))𝑐𝑧+^∞_𝑛=2(1−𝛾)(1−2𝜆)(1−𝑐)𝜇_𝑛(1+(𝑛−1)𝜆)[𝑛(𝑘+1)+(𝑘+𝛾)]Γ_𝑛𝛼₁𝑧^𝑛.(2.16) Since ^∞_𝑛=2(1−𝛾)(1−2𝜆)(1−𝑐)𝜇_𝑛(1+(𝑛−1)𝜆)[𝑛(𝑘+1)+(𝑘+𝛾)]Γ_𝑛𝛼₁(1+(𝑛−1)𝜆)[𝑛(𝑘+1)+(𝑘+𝛾)]Γ^𝑛𝛼₁=(1−𝛾)(1−2𝜆)(1−𝑐)_∞_𝑛=2𝜇_𝑛=1−𝜇₁≤1,(2.17) it follows from Theorem 1.5 that the function 𝑓∈Σ_𝑃(𝛾,𝑘,𝜆,𝑐). Conversely, suppose that 𝑓∈Σ_𝑃(𝛾,𝑘,𝜆,𝑐). Since 𝑎_𝑛≤(1−𝛾)(1−2𝜆)(1−𝑐)(1+(𝑛−1)𝜆)[𝑛(𝑘+1)+(𝑘+𝛾)]Γ_𝑛𝛼₁,𝑛≥2.(2.18) Setting 𝜇_𝑛=(1+(𝑛−1)𝜆)[𝑛(𝑘+1)+(𝑘+𝛾)]Γ^𝑛𝛼₁(1−𝛾)(1−2𝜆)(1−𝑐)𝑎^𝑛𝜇₁,=1−^∞_𝑛=2𝜇_𝑛.(2.19) it follows that 𝑓(𝑧)=^∞_𝑛=1𝜇_𝑛𝑓_𝑛(𝑧).(2.20) This completes the proof of the theorem.

Theorem 2.6. The class Σ_𝑃(𝛾,𝑘,𝜆,𝑐) is closed under linear combination.

Proof. Suppose that the function 𝑓 be given by (1.20), and let the function 𝑔 be given by 𝑔(𝑧)=1𝑧+(1(−2𝑘𝛾)+(1𝛾−+21𝜆))𝑐𝑧+^∞_𝑛=2||𝑏_𝑛||𝑧^𝑛,𝑛≥2.(2.21) Assuming that 𝑓 and 𝑔 are in the class Σ_𝑃(𝛾,𝑘,𝜆,𝑐), it is enough to prove that the function ℎ defined by ℎ(𝑧)=𝜇𝑓(𝑧)+(1−𝜇)𝑔(𝑧),0≤𝜇≤1,(2.22) is also in the class Σ_𝑃(𝛾,𝑘,𝜆,𝑐). Since ℎ(𝑧)=1𝑧+(1(−2𝑘𝛾)+(1𝛾−+21𝜆))𝑐𝑧+^∞_𝑛=2||𝑎_𝑛𝜇+(1−𝜇)𝑏_𝑛||𝑧^𝑛,(2.23) we observe that ^∞_𝑛=2(1+(𝑛−1)𝜆)[𝑛(𝑘+1)+(𝑘+𝛾)]Γ_𝑛𝛼₁||𝑎_𝑛𝜇+(1−𝜇)𝑏_𝑛||≤(1−𝛾)(1−2𝜆)(1−𝑐),(2.24) with the aid of Theorem 2.1. Thus, ℎ∈Σ_𝑃(𝛾,𝑘,𝜆,𝑐).

Next we determine the radii of meromorphically starlikeness and convexity of order 𝛿 for functions in the class Σ_𝑃(𝛾,𝑘,𝜆,𝑐).

Theorem 2.7. Let the function 𝑓(𝑧) defined by (1.20) be in the class Σ_𝑃(𝛾,𝑘,𝜆,𝑐), then (i)𝑓 is meromorphically starlike of order 𝛿(0≤𝛿<1) in the disk |𝑧|<𝑟₁(𝛾,𝑘,𝜆,𝑐,𝛿) where 𝑟₁(𝛾,𝑘,𝜆,𝑐,𝛿), is the largest value for which (3−𝛿)(1−𝛾)(1−2𝜆)𝑐(2𝑘+𝛾+1)𝑟²+(𝑛+2−𝛿)(1−𝛾)(1−2𝜆)(1−𝑐)(1+(𝑛−1)𝜆)[𝑛(𝑘+1)+(𝑘+𝛾)]𝑟^𝑛+1≤(1−𝛿),𝑛≥2,(2.25)(ii)𝑓 is meromorphically convex of order 𝛿(0≤𝛿<1) in the disk |𝑧|<𝑟₂(𝛾,𝑘,𝜆,𝑐,𝛿) where 𝑟₂(𝛾,𝑘,𝜆,𝑐,𝛿), is the largest value for which(3−𝛿)(1−𝛾)(1−2𝜆)𝑐(2𝑘+𝛾+1)𝑟²+𝑛(𝑛+2−𝛿)(1−𝛾)(1−2𝜆)(1−𝑐)(1+(𝑛−1)𝜆)[𝑛(𝑘+1)+(𝑘+𝛾)]𝑟^𝑛+1≤(1−𝛿),𝑛≥2.(2.26) Each of these results is sharp for function 𝑓(𝑧) given by (2.3).

Proof. It is enough to highlight that ||||𝑧𝑓^(𝑧)𝑓(𝑧)+1||||≤1−𝛿,|𝑧|<𝑟¹.(2.27) Thus, we have ||||𝑧𝑓^(𝑧)=𝑓||||−(𝑧1)/𝑧++1((||||𝑉)(𝐸)𝑐/(𝑄))𝑧+∑_∞𝑛=2𝑛Γ_𝑛𝛼₁𝑎_𝑛𝑧^𝑛+1/𝑧+((𝑉)(𝐸)𝑐/(𝑄))𝑧+∑^∞𝑛=2Γ_𝑛𝛼₁𝑎_𝑛𝑧^𝑛1/𝑧+((𝑉)(𝐸)𝑐/(𝑄))𝑧+∑^∞𝑛=2Γ_𝑛𝛼₁𝑎_𝑛𝑧^𝑛||||,(2.28) where (𝑄) denotes (2𝑘+𝛾+1),𝑉 denotes (1−𝛾), and (𝐸) denotes (1−2𝜆).
Hence (2.28) holds true if 2(1−𝛾)(1−2𝜆)𝑐(2𝑘+𝛾+1)𝑟²+^∞_𝑛=2(𝑛+1)Γ_𝑛𝛼₁𝑎_𝑛𝑟^𝑛+1≤(1−𝛿)1−(1−𝛾)(1−2𝜆)𝑐(2𝑘+𝛾+1)𝑟²−^∞_𝑛=2Γ_𝑛𝛼₁𝑎_𝑛𝑟^𝑛+1,(2.29) or, (3−𝛿)(1−𝛾)(1−2𝜆)𝑐(2𝑘+𝛾+1)𝑟²+^∞_𝑛=2(𝑛+2−𝛿)Γ_𝑛𝛼₁𝑎_𝑛𝑟^𝑛+1≤(1−𝛿),(2.30) and it follows that, from (2.1), we may take 𝑎_𝑛≤(1−𝛾)(1−2𝜆)(1−𝑐)(1+(𝑛−1)𝜆)[𝑛(𝑘+1)+(𝑘+𝛾)]Γ_𝑛𝛼₁𝜇^𝑛,𝑛≥2,(2.31) where 𝜇_𝑛≥0 and ∑^∞𝑛=2𝜇_𝑛≤1.
For each fixed 𝑟, we choose the positive integer 𝑛₀=𝑛₀(𝑟) for which (𝑛+2−𝛿)(1+(𝑛−1)𝜆)[𝑛(𝑘+1)+(𝑘+𝛾)]Γ^𝑛𝛼₁𝑟^𝑛+1(2.32) is maximal. Then it follows that ^∞_𝑛=2(𝑛+2−𝛿)Γ_𝑛𝛼₁𝑎_𝑛𝑟^𝑛+1≤𝑛⁰+2−𝛿(1−𝛾)(1−2𝜆)(1−𝑐)1+𝑛₀−1𝜆𝑛₀(𝑘+1)+(𝑘+𝛾)𝑟^𝑛0+1.(2.33) Then 𝑓 is starlike of order 𝛿 in 0<|𝑧|<𝑟₁(𝛾,𝑘,𝜆,𝑐,𝛿) provided that (3−𝛿)(1−𝛾)(1−2𝜆)𝑐(2𝑘+𝛾+1)𝑟²+𝑛⁰+2−𝛿(1−𝛾)(1−2𝜆)(1−𝑐)1+𝑛₀−1𝜆𝑛₀(𝑘+1)+(𝑘+𝛾)𝑟^𝑛0+1≤(1−𝛿).(2.34) We find the value 𝑟₀=𝑟₀(𝑘,𝑐,𝛿,𝑛) and the corresponding integer 𝑛₀(𝑟₀) so that (3−𝛿)(1−𝛾)(1−2𝜆)𝑐(2𝑘+𝛾+1)𝑟²+𝑛⁰+2−𝛿(1−𝛾)(1−2𝜆)(1−𝑐)1+𝑛₀−1𝜆𝑛₀(𝑘+1)+(𝑘+𝛾)𝑟^𝑛0+1=(1−𝛿).(2.35) It is the value for which the function 𝑓(𝑧) is starlike in 0<|𝑧|<𝑟₀. (ii) In a similar manner, we can prove our result providing the radius of meromorphic convexity of order 𝛿(0≤𝛿<1) for functions in the class Σ_𝑃(𝛾,𝑘,𝜆,𝑐), so we skip the details of the proof of (ii).