Subclasses of Starlike Functions Associated with Fractional 𝑞-Calculus Operators

Abstract

Making use of fractional 𝑞-calculus operators, we introduce a new subclass ℳ_𝑞(𝜆,𝛾,𝑘) of starlike functions and determine the coefficient estimate, extreme points, closure theorem, and distortion bounds for functions in ℳ_𝑞(𝜆,𝛾,𝑘). Furthermore we discuss neighborhood results, subordination theorem, partial sums, and integral means inequalities for functions in ℳ_𝑞(𝜆,𝛾,𝑘).

1. Introduction and Preliminaries

Denote by 𝒜 the class of functions of the form 𝑓(𝑧)=𝑧+^∞_𝑛=2𝑎_𝑛𝑧^𝑛 (1) which are analytic and univalent in the open disc 𝑈={𝑧∶|𝑧|<1} and normalized by 𝑓(0)=0=𝑓^′(0)−1. Due to Silverman [1], denote by 𝒯 a subclass of 𝒜 consisting of functions of the form 𝑓(𝑧)=𝑧−^∞_𝑛=2𝑎_𝑛𝑧^𝑛,𝑎_𝑛≥0,𝑧∈𝑈. (2)

The fractional calculus operator has gained importance and popularly due to vast potential demonstrated applications in various fields of science, engineering and also in the geometric function theory. The fractional 𝑞-calculus operator is the extension of the ordinary fractional calculus in the 𝑞-theory. Recently Purohit and Raina [2] investigated applications of fractional 𝑞-calculus operator to define new classes of functions which are analytic in the open unit disc. We recall the definitions of fractional 𝑞-calculus operators of complex valued function 𝑓(𝑧).

The 𝑞-shifted factorial is defined for 𝛼,𝑞∈ℂ as a product of 𝑛 factors by 𝛼;𝑞_𝑛=(⋯1,𝑛=0,1−𝛼)1−𝛼𝑞1−𝛼𝑞^𝑛−1,𝑛∈𝑁, (3) and in terms of basic analogue of the gamma function 𝑞^𝛼;𝑞_𝑛=Γ_𝑞(𝛼+𝑛)1−𝑞^𝑛Γ_𝑞(𝛼),𝑛>0. (4) Due to Gasper and Rahman [3], the recurrence relation for 𝑞-gamma function is given by Γ_𝑞(1+𝛼)=1−𝑞^𝛼Γ_𝑞(𝛼)1−𝑞 (5) and the 𝑞 binomial expansion is given by 𝑥−𝑦_𝑣=𝑥^𝑣−𝑦𝑥;𝑞=𝑥^𝑣∞_𝑛=01−𝑦/𝑥𝑞^𝑛1−𝑦/𝑥𝑞^𝑣+𝑛=𝑥^𝑣₁Φ₀𝑞^−𝑣;−;𝑞,𝑦𝑞^𝑣𝑥. (6) Further the 𝑞-derivative and 𝑞-integral of functions 𝑓 defined on the subset of ℂ are, respectively, given by 𝐷_𝑞,𝑧𝑓(𝑧)=𝑓(𝑧)−𝑓𝑧𝑞𝑧,,1−𝑞𝑧≠0,𝑞≠0_𝑧0𝑓(𝑡)𝑑𝑡;𝑞=𝑧1−𝑞^∞_𝑘=0𝑞^𝑘𝑓𝑧𝑞^𝑘. (7) It is interest to note that lim_𝑞→1⁻((𝑞^𝛼;𝑞)_𝑛/(1−𝑞)^𝑛)=(𝛼)_𝑛=𝛼(𝛼+1)⋯(𝛼+𝑛−1) the familiar Pochhammer symbol. Due to Kim and Srivastava [4], we recall the following definitions of fractional 𝑞-integral and fractional 𝑞-derivative operators, which are very much useful for our study.

Definition 1. Let the function 𝑓(𝑧) be analytic in a simply connected region of the 𝑧-plane containing the origin. The fractional 𝑞-integral of 𝑓 of order 𝜇 is defined by ℐ^𝜇_𝑞,𝑧𝑓(𝑧)=𝒟^−𝜇_𝑞,𝑧𝑓(𝑧)=1Γ_𝑞𝜇^𝑧₀𝑧−𝑞𝑡_𝜇−1𝑓(𝑡)𝑑𝑡;𝑞,𝜇>0, (8) where (𝑧−𝑡𝑞)_𝜇−1 can be expressed as the 𝑞-binomial given by (6) and the series ₁Φ₀[𝜇;−;𝑞,𝑧] is a single valued when |arg(𝑧)|<𝜋 and |𝑧|<1, therefore the function (𝑧−𝑡𝑞)_𝜇−1 in (8) is single valued when |arg(−𝑡𝑞^𝜇/𝑧)|<𝜋, |𝑡𝑞^𝜇|<1 and |arg(𝑧)|<𝜋.

Definition 2. The fractional 𝑞-derivative operator of order 𝜇 is defined for a function 𝑓(𝑧) by 𝒟^𝜇_𝑞,𝑧𝑓(𝑧)=𝒟_𝑞,𝑧ℐ^1−𝜇_𝑞,𝑧𝑓(𝑧)=1Γ_𝑞𝐷1−𝜇^𝑞,𝑧×^𝑧₀𝑧−𝑞𝑡_−𝜇𝑓(𝑡)𝑑𝑡;𝑞,0≤𝜇<1, (9) where the function 𝑓(𝑧) is constrained, and the multiplicity of the function (𝑧−𝑡𝑞)_−𝜇 is removed as in Definition 1.

Definition 3. Under the hypothesis of Definition 2, the fractional derivative of order 𝜇 is defined by 𝐷^𝜇_𝑞,𝑧𝑓(𝑧)=𝐷^𝑚_𝑞,𝑧ℐ^𝑚−𝜇_𝑞,𝑧𝑓(𝑧),𝑚−1≤𝜇<𝑚;𝑚∈𝑁₀. (10)

With the aid of the above definitions, and their known extensions involving 𝑞-differintegral operator we define the linear operator Ω^𝜇_𝑞,𝑧𝑓(𝑧)∶𝒯→𝒯Ω^𝜇_𝑞,𝑧𝑓(𝑧)=Γ_𝑞2−𝜇Γ_𝑞(2)𝑧^𝜇−1𝐷^𝜇_𝑞,𝑧𝑓(𝑧)=𝑧−^∞_𝑛=2Φ_𝑞𝑛,𝜇𝑎_𝑛𝑧^𝑛, (11) where Φ_𝑞=Γ𝑛,𝜇^𝑞2−𝜇Γ_𝑞(𝑛+1)Γ_𝑞(2)Γ_𝑞,𝑛+1−𝜇 (12) where 𝜇<2, 0<𝑞<1 and 𝑧∈𝑈. Here 𝐷^𝜇_𝑞,𝑧𝑓(𝑧) in (10) represents, respectively, a fractional 𝑞-integral of 𝑓(𝑧) of order 𝜇 when −∞<𝜇<0 and fractional 𝑞-derivative of 𝑓(𝑧) of order 𝜇 when 0≤𝜇<2.

In 1975, Silverman [1] studied two interesting subclasses of 𝑓(𝑧)∈𝒯, namely, 𝒯^∗(𝛾), the class of starlike function of order 𝛾 (0≤𝛾<1) if ℜ(𝑧𝑓^′(𝑧)/𝑓(𝑧))>𝛾 and 𝐶(𝛾), the class of convex function of order 𝛾 (0≤𝛾<1) if ℜ(1+(𝑧𝑓^′′(𝑧)/𝑓^′(𝑧)))>𝛾. Motivated by the earlier works of Goodman [5] and Rønning [6, 7] in this paper we define the following new subclass of 𝑘-starlike functions of order 𝛾 based on the 𝑞-fractional operator.

For 0≤𝜆<1, 0≤𝛾<1, 𝜇<2, and 𝑘≥0, we let ℳ_𝑞(𝜆,𝛾,𝑘) be the subclass of 𝑇 consisting of functions of the form (2) and satisfying the analytic criterion ℜΩ^𝜇_𝑞,𝑧𝑓(𝑧)^′+𝜆𝑧Ω^𝜇_𝑞,𝑧𝑓(𝑧)^′′Ω^𝜇_𝑞,𝑧𝑓(𝑧)^′+𝑧Ω^𝜇_𝑞,𝑧𝑓(𝑧)^′′Ω−𝛾>𝑘^{𝜇𝑞,𝑧}𝑓(𝑧)^′+𝜆𝑧Ω^𝜇_𝑞,𝑧𝑓(𝑧)^′′Ω^𝜇_𝑞,𝑧𝑓(𝑧)^′+𝑧Ω^𝜇_𝑞,𝑧𝑓(𝑧)^′′−1 (13)𝑧∈𝑈, and Ω^𝜇_𝑞,𝑧𝑓(𝑧) is given by (11).

For different choices of 𝜆 we state some special cases of subclasses of ℳ_𝑞(𝜆,𝛾,𝑘) as illustrated in the following examples.

Example 4. For 𝜆=0, we let 𝒰𝒞ℳ_𝑞Ω𝛾,𝑘∶=𝑓∈𝒯∶ℜ^{𝜇𝑞,𝑧}𝑓(𝑧)^′Ω^𝜇_𝑞,𝑧𝑓(𝑧)^′+𝑧Ω^𝜇_𝑞,𝑧𝑓(𝑧)^′′Ω−𝛾>𝑘^{𝜇𝑞,𝑧}𝑓(𝑧)^′Ω^𝜇_𝑞,𝑧𝑓(𝑧)^′+𝑧Ω^𝜇_𝑞,𝑧𝑓(𝑧)^′′−1, (14) where 0≤𝛾<1, 𝜇<2, 𝑘≥0, and 𝑧∈𝑈.

Example 5. For 𝜆=0 and 𝑘=0, we let ℳ𝒯_𝑞𝛾Ω∶=𝑓∈𝒯∶ℜ^{𝜇𝑞,𝑧}𝑓(𝑧)^′Ω^𝜇_𝑞,𝑧𝑓(𝑧)^′+𝑧Ω^𝜇_𝑞,𝑧𝑓(𝑧)^′′,>𝛾,𝑧∈𝑈 (15) where 0≤𝛾<1, 𝜇<2 and 𝑧∈𝑈.

In this paper we determine the coefficient estimate, extreme points, closure theorem, and distortion bounds for functions in ℳ_𝑞(𝜆,𝛾,𝑘). Furthermore we discuss neighborhood results, subordination theorem, partial sums, and integral means inequalities for functions in ℳ_𝑞(𝜆,𝛾,𝑘).

2. Characterization Properties of ℳ_𝑞(𝜆,𝛾,𝑘)

We recall the following lemmas to prove our main results.

Lemma 6. If 𝛾 is a real number and 𝑤 is a complex number, then ℜ(𝑤)≥𝛾⇔|𝑤+(1−𝛾)|−|𝑤−(1+𝛾)|≥0.

Lemma 7. If 𝑤 is a complex number and 𝛾, 𝑘 are real numbers, then ℜ(𝑤)≥𝑘|𝑤−1|+𝛾⟺ℜ𝑤1+𝑘𝑒^𝑖𝜃−𝑘𝑒^𝑖𝜃≥𝛾,−𝜋≤𝜃≤𝜋. (16)

Lemma 8. Let 0≤𝜆<1, 0≤𝛾<1, 𝑘≥0 then a function 𝑓∈ℳ_𝑞(𝜆,𝛾,𝑘) if and only if ^∞_𝑛=2𝑛(1+𝑛𝜆−𝜆)(1+𝑘)−𝑛𝛾+𝑘Φ_𝑞𝑛,𝜇𝑎_𝑛≤1−𝛾, (17) where Φ_𝑞(𝑛,𝜇) is given by (12).

Proof. Let a function 𝑓 of the form (2) in 𝑇 satisfy the condition (17). We will show that (13) is satisfied, and so 𝑓∈ℳ_𝑞(𝜆,𝛾,𝑘). Using Lemma 7, it is enough to show that ℜΩ^𝜇_𝑞,𝑧𝑓(𝑧)^′+𝜆𝑧Ω^𝜇_𝑞,𝑧𝑓(𝑧)^′′Ω^𝜇_𝑞,𝑧𝑓(𝑧)^′+𝑧Ω^𝜇_𝑞,𝑧𝑓(𝑧)^′′1+𝑘𝑒^𝑖𝜃−𝑘𝑒^𝑖𝜃>𝛾,−𝜋≤𝜃≤𝜋. (18) That is, suppose 𝑓∈ℳ_𝑞(𝜆,𝛾,𝑘), then by Lemma 7 and by choosing the values of 𝑧 on the positive real axis inequality (18) reduces to ℜ−1−𝛾_∞𝑛=2𝑛1+𝑘𝑒^𝑖𝜃(1+𝜆𝑛−𝜆)−𝑛𝛾+𝑘𝑒^𝑖𝜃×Φ_𝑞𝑛,𝜇𝑎_𝑛𝑧^𝑛−1×1−^∞_𝑛=2𝑛²Φ_𝑞𝑛,𝜇𝑎_𝑛𝑧^𝑛−1−1≥0. (19) Since Re(−𝑒^𝑖𝜃)≥−𝑒^𝑖0=−1, the previous inequality reduces to ℜ−1−𝛾_∞𝑛=2𝑛(𝑘+1)(1+𝑛𝜆−𝜆)−𝑛𝛾+𝑘×Φ_𝑞𝑛,𝜇𝑎_𝑛𝑟^𝑛−1×1−^∞_𝑛=2𝑛²Φ_𝑞𝑛,𝜇𝑎_𝑛𝑟^𝑛−1−1≥0. (20) Letting 𝑟→1⁻ and by the mean value theorem we get desired inequality (17).
Conversely, let (17) hold; we will show that (13) is satisfied, and so 𝑓∈ℳ_𝑞(𝜆,𝛾,𝑘). In view of Lemma 6, ℜ(𝑤)>𝛾⇔|𝑤−(1+𝛾)|<|𝑤+(1−𝛾)|, it is enough to show that 𝐴(𝑧)𝐵(𝑧)−𝐴(𝑧)1+𝑘𝐵(𝑧)<𝐴(𝑧)−1+𝛾𝐵(𝑧)+𝐴(𝑧)1−𝑘𝐵(𝑧)−1−𝛾, (21) where 𝐴(𝑧)∶=Ω^𝜇_𝑞,𝑧𝑓(𝑧)^′+𝜆𝑧Ω^𝜇_𝑞,𝑧𝑓(𝑧)^′′=1−^∞_𝑛=2𝑛(1+𝑛𝜆−𝜆)Φ_𝑞𝑛,𝜇𝑎_𝑛𝑧^𝑛−1,𝐵(𝑧)∶=Ω^𝜇_𝑞,𝑧𝑓(𝑧)^′+𝑧Ω^𝜇_𝑞,𝑧𝑓(𝑧)^′′=1−^∞_𝑛=2𝑛²Φ_𝑞𝑛,𝜇𝑎_𝑛𝑧^𝑛−1. (22) Hence, one has 𝐴(𝑧)𝐿=𝐵(𝑧)−𝐴(𝑧)1+𝑘𝐵(𝑧)<1−1+𝛾|𝐵(𝑧×)|−𝛾−^∞_𝑛=2𝑛(1+𝑛𝜆−𝜆)(1+𝑘)−𝑛²1+𝛾+𝑘×Φ_𝑞𝑛,𝜇𝑎_𝑛𝑧^𝑛−1<1|𝐵(𝑧−)|2−𝛾_∞𝑛=2𝑛(1+𝑛𝜆−𝜆)(1+𝑘)+𝑛²1−𝛾−𝑘×Φ_𝑞𝑛,𝜇𝑎_𝑛𝑧^𝑛−1𝐴(𝑧)<𝑅=𝐵(𝑧)+𝐴(𝑧)1−𝑘𝐵(𝑧)−1−𝛾, (23) and it is easy to show that 𝑅−𝐿>0, by the given condition (17), and the proof is complete.

Corollary 9. If 𝑓∈ℳ_𝑞(𝜆,𝛾,𝑘), then |𝑎_𝑛|≤(1−𝛾)/Ψ(𝜆,𝛾,𝑘,𝑛), 0≤𝜆≤1, 0≤𝛾<1, 𝑘≥0, where Ψ(𝜆,𝛾,𝑘,𝑛)=𝑛[(1+𝑛𝜆−𝜆)(1+𝑘)−(𝛾+𝑘)]Φ_𝑞(𝑛,𝜇). Equality holds for the function 𝑓(𝑧)=𝑧−((1−𝛾)/Ψ(𝜆,𝛾,𝑘,𝑛))𝑧^𝑛.

For the sake of brevity we let (i)Ψ)Φ𝜆,𝛾,𝑘,𝑛=𝑛(1+𝑛𝜆−𝜆)(1+𝑘−𝑛𝛾+𝑘_𝑞𝑛,𝜇,(ii)Ψ𝜆,𝛾,𝑘,2=2(1+𝜆)(𝑘+1)−2𝛾+𝑘Φ_𝑞2,𝜇,(iii)Φ_𝑞=Γ2,𝜇^𝑞2−𝜇Γ_𝑞(3)Γ_𝑞(2)Γ_𝑞=3−𝜇1−𝑞²1−𝑞^2−𝜇, (24)(25)(26) unless otherwise stated.

Remark 10. First we show that the function Φ_𝑞=Γ𝑛,𝜇^𝑞2−𝜇Γ_𝑞(𝑛+1)Γ_𝑞(2)Γ_𝑞𝑛+1−𝜇,𝑛≥2 (27) is a decreasing function of 𝑛 for 𝜇<2, 0<𝑞<1. It follows that Φ_𝑞𝑛+1,𝜇Φ_𝑞=Γ𝑛,𝜇^𝑞𝑛+1−𝜇Γ_𝑞(𝑛+2)Γ_𝑞(𝑛+1)Γ_𝑞𝑛+2−𝜇 (28) and it is sufficient to consider here the value 𝑛<2, so that on using (5) we get Φ_𝑞3,𝜇Φ_𝑞=2,𝜇1−𝑞³1−𝑞^3−𝜇,0<𝑞<1. (29) The function Φ_𝑞(𝑛,𝜇) is a decreasing function of 𝑛 if Φ_𝑞(3,𝜇)/Φ_𝑞(2,𝜇)≤1, and this gives (1−𝑞³)/(1−𝑞^3−𝜇)≤1. Multiplying the previous inequality both sides by 1−𝑞^3−𝜇 provided 𝜇<2; we are at once lead to the inequality 𝜇≤0. Thus, Φ_𝑞(𝑛,𝜇) is a decreasing function of 𝑛 for −∞<𝜇<2, 0<𝑞<1.

Now by routine procedure using the techniques employed by Silverman [1] we can easily prove the following theorems.

Theorem 11. Let the function 𝑓(𝑧) defined by (2) belong to ℳ_𝑞(𝜆,𝛾,𝑘), then 𝑟−1−𝛾Ψ𝑟𝜆,𝛾,𝑘,2²≤𝑓(𝑧)≤𝑟+1−𝛾Ψ𝑟𝜆,𝛾,𝑘,2²,|𝑧|2=𝑟,1−1−𝛾Ψ𝑟≤𝑓𝜆,𝛾,𝑘,2_′(𝑧2)≤1+1−𝛾Ψ|𝑧|𝜆,𝛾,𝑘,2𝑟,=𝑟. (30) Equalities are sharp for the function 𝑓(𝑧)=𝑧−((1−𝛾)/Ψ(𝜆,𝛾,𝑘,2))𝑧², where Ψ(𝜆,𝛾,𝑘,2) is obtained from (25).

Theorem 12 (extreme points). The extreme points of ℳ_𝑞(𝜆,𝛾,𝑘) are 𝑓₁(𝑧)=𝑧 and 𝑓_𝑛(𝑧)=𝑧−((1−𝛾)/Ψ(𝜆,𝛾,𝑘,𝑛))𝑧^𝑛, for 𝑛=2,3,4,…. Then 𝑓∈ℳ_𝑞(𝜆,𝛾,𝑘) if and only if it can be expressed in the form 𝑓(𝑧)=∑^∞_𝑛=1𝜔_𝑛𝑓_𝑛(𝑧), (𝜔_𝑛≥0), ∑^∞_𝑛=1𝜔_𝑛=1, where Ψ(𝜆,𝛾,𝑘,𝑛) is defined in (24).

Theorem 13. Let the functions 𝑓_𝑗(𝑧) (𝑗=1,2,…,𝑚) defined by 𝑓_𝑗(𝑧)=𝑧−^∞_𝑛=2𝑎_𝑛,𝑗𝑧^𝑛𝑓𝑜𝑟𝑎_𝑛,𝑗≥0,𝑧∈𝑈. (31) be in the classes ℳ_𝑞(𝜆,𝛾_𝑗,𝑘) (𝑗=1,2,…,𝑚), respectively. Then the function ℎ(𝑧)=𝑧−(1/𝑚)∑^∞_𝑛=2(∑^𝑚_𝑗=1𝑎_𝑛,𝑗)𝑧^𝑛 is in the class ℳ_𝑞(𝜆,𝛾,𝑘), where 𝛾=min_{1≤𝑗≤𝑚}{𝛾_𝑗} (−1≤𝛾_𝑗<1).

3. Neighbourhood Results

In this section, we discuss neighbourhood results of the class ℳ_𝑞(𝜆,𝛾,𝑘). Following [8, 9], we define the 𝛿-neighbourhood of function 𝑓(𝑧)∈𝒯 by 𝑁_𝛿𝑓∶=ℎ∈𝒯∶ℎ(𝑧)=𝑧−^∞_𝑛=2𝑑_𝑛𝑧^𝑛,_∞𝑛=2𝑛𝑎_𝑛−𝑑_𝑛≤𝛿. (32) Particularly for the identity function 𝑒(𝑧)=𝑧, we have 𝑁_𝛿(𝑒)∶=ℎ∈𝒯∶ℎ(𝑧)=𝑧−^∞_𝑛=2𝑑_𝑛𝑧^𝑛,_∞𝑛=2𝑛𝑑_𝑛≤𝛿. (33)

Theorem 14. If 𝛿∶=1−𝛾(1+𝜆)(1+𝑘)−2𝛾+𝑘Φ_𝑞2,𝜇 (34) then ℳ_𝑞(𝜆,𝛾,𝑘)⊂𝑁_𝛿(𝑒), where Ψ(𝜆,𝛾,𝑘,2) is defined in (25).

Proof. For 𝑓∈ℳ_𝑞(𝜆,𝛾,𝑘), Lemma 8 immediately yields Ψ𝜆,𝛾,𝑘,2^∞_𝑛=2𝑎_𝑛≤1−𝛾, (35) so that ^∞_𝑛=2𝑎_𝑛≤1−𝛾Ψ.𝜆,𝛾,𝑘,2 (36) On the other hand, we find from (17) and (36) that ^∞_𝑛=2𝑛𝑎_𝑛≤1−𝛾(1+𝜆)(1+𝑘)−2𝛾+𝑘Φ_𝑞,2,𝜇 (37) which, in view of the definition (33), proves Theorem 14.

Now we determine the neighborhood for the class ℳ_𝑞(𝜌,𝜆,𝛾,𝑘) which we define as follows. A function 𝑓∈𝒯 is said to be in the class ℳ_𝑞(𝜌,𝜆,𝛾,𝑘) if there exists a function ℎ∈ℳ_𝑞(𝜌,𝜆,𝛾,𝑘) such that 𝑓(𝑧)ℎ(𝑧)−1<1−𝜌,𝑧∈𝑈,0≤𝜌<1. (38)

Theorem 15. If ℎ∈ℳ_𝑞(𝜌,𝜆,𝛾,𝑘) and 𝜌=1−𝛿Ψ𝜆,𝛾,𝑘,22Ψ𝜆,𝛾,𝑘,2−1−𝛾 (39) then 𝑁_𝛿(ℎ)⊂ℳ_𝑞𝜌,𝜆,𝛾,𝑘, (40) where Ψ(𝜆,𝛾,𝑘,2) is defined in (25).

Proof. Suppose that 𝑓∈𝑁_𝛿(ℎ); we then find from (32) that ^∞_𝑛=2𝑛𝑎_𝑛−𝑑_𝑛≤𝛿 (41) which yields ^∞_𝑛=2𝑎_𝑛−𝑑_𝑛≤𝛿2. (42) Next, since ℎ∈ℳ_𝑞(𝜆,𝛾,𝑘), we have ^∞_𝑛=2𝑑_𝑛=1−𝛾Ψ,𝜆,𝛾,𝑘,2 (43) so that 𝑓(𝑧)ℎ(𝑧)<∑−1^∞𝑛=2𝑎_𝑛−𝑑_𝑛1−∑^∞_𝑛=2𝑑_𝑛≤𝛿2×Ψ𝜆,𝛾,𝑘,2Ψ−≤𝜆,𝛾,𝑘,21−𝛾𝛿Ψ𝜆,𝛾,𝑘,22Ψ𝜆,𝛾,𝑘,2−1−𝛾=1−𝜌 (44) provided that 𝜌 is given precisely by (39). Thus 𝑓∈ℳ_𝑞(𝜌,𝜆,𝛾,𝑘) for 𝜌 given by (39), completes the proof.

4. Subordination Results

Now we recall the following results due to Wilf [10], which are very much needed for our study.

Definition 16 (subordination principle [11]). For analytic functions 𝑔 and ℎ with 𝑔(0)=ℎ(0), 𝑔 is said to be subordinate to ℎ, denoted by 𝑔≺ℎ, if there exists an analytic function 𝑤 such that 𝑤(0)=0, |𝑤(𝑧)|<1, and 𝑔(𝑧)=ℎ(𝑤(𝑧)), for all 𝑧∈𝑈.

Definition 17 (subordinating factor sequence). A sequence {𝑏_𝑛}^∞_𝑛=1 of complex numbers is said to be a subordinating sequence if, whenever 𝑓(𝑧)=∑^∞_𝑛=1𝑎_𝑛𝑧^𝑛, 𝑎₁=1 is regular, univalent, and convex in 𝑈, we have ^∞_𝑛=1𝑏_𝑛𝑎_𝑛𝑧^𝑛≺𝑓(𝑧),𝑧∈𝑈. (45)

Lemma 18. The sequence {𝑏_𝑛}^∞_𝑛=1 is a subordinating factor sequence if and only if Re1+2^∞_𝑛=1𝑏_𝑛𝑧^𝑛>0,𝑧∈𝑈. (46)

Theorem 19. Let 𝑓∈ℳ_𝑞(𝜆,𝛾,𝑘) and 𝑔(𝑧) be any function in the usual class of convex functions 𝐶, then Ψ𝜆,𝛾,𝑘,22(𝑧)(𝑧),1−𝛾+Ψ𝜆,𝛾,𝑘,2𝑓∗𝑔≺𝑔 (47) where 0≤𝛾<1, 𝑘≥0, 0≤𝜆≤1, and Re𝑓(𝑧)>−1−𝛾+Ψ𝜆,𝛾,𝑘,2Ψ𝜆,𝛾,𝑘,2,𝑧∈𝑈. (48) The constant factor Ψ(𝜆,𝛾,𝑘,2)/2[1−𝛾+Ψ(𝜆,𝛾,𝑘,2)] in (47) cannot be replaced by a larger number.

Proof. Let 𝑓∈ℳ_𝑞(𝜆,𝛾,𝑘), and suppose that 𝑔(𝑧)=𝑧+∑^∞_𝑛=2𝑏_𝑛𝑧^𝑛∈𝐶. Then Ψ𝜆,𝛾,𝑘,22(𝑧)=Ψ1−𝛾+Ψ𝜆,𝛾,𝑘,2𝑓∗𝑔𝜆,𝛾,𝑘,221−𝛾+Ψ𝜆,𝛾,𝑘,2𝑧+^∞_𝑛=2𝑏_𝑛𝑎_𝑛𝑧^𝑛. (49) Thus, by Definition 17, the subordination result holds true if Ψ𝜆,𝛾,𝑘,221−𝛾+Ψ𝜆,𝛾,𝑘,2^∞_𝑛=1 (50) is a subordinating factor sequence, with 𝑎₁=1. In view of Lemma 18, this is equivalent to the following inequality: Re1+^∞_𝑛=1Ψ𝜆,𝛾,𝑘,2𝑎1−𝛾+Ψ𝜆,𝛾,𝑘,2^𝑛𝑧^𝑛>0,𝑧∈𝑈. (51) By noting the fact that Ψ(𝜆,𝛾,𝑘,𝑛)/(1−𝛾) is increasing function for 𝑛≥2 and in particular Ψ𝜆,𝛾,𝑘,2≤Ψ1−𝛾𝜆,𝛾,𝑘,𝑛1−𝛾,𝑛≥2, (52) therefore, for |𝑧|=𝑟<1, we have ΨRe1+𝜆,𝛾,𝑘,21−𝛾+Ψ𝜆,𝛾,𝑘,2^∞_𝑛=1𝑎_𝑛𝑧^𝑛Ψ=Re1+𝜆,𝛾,𝑘,2𝑧+∑1−𝛾+Ψ𝜆,𝛾,𝑘,2_∞𝑛=2Ψ𝜆,𝛾,𝑘,2𝑎_𝑛𝑧^𝑛Ψ1−𝛾+Ψ𝜆,𝛾,𝑘,2≥1−𝜆,𝛾,𝑘,2𝑟−∑1−𝛾+Ψ𝜆,𝛾,𝑘,2_∞𝑛=2Ψ𝜆,𝛾,𝑘,𝑛𝑎_𝑛𝑟^𝑛|𝑧|1−𝛾+Ψ𝜆,𝛾,𝑘,2>0,=𝑟<1, (53) where we have also made use of the assertion (17) of Lemma 8. This evidently proves the inequality (51) and hence also the subordination result (47) asserted by Lemma 8.
The inequality (48) follows from (47) by taking 𝑔(𝑧)=𝑧1−𝑧=𝑧+^∞_𝑛=2𝑧^𝑛∈𝐶. (54) Next we consider the function 𝐹(𝑧)∶=𝑧−1−𝛾Ψ𝑧𝜆,𝛾,𝑘,2², (55) where 0≤𝛾<1, 𝑘≥0, 0≤𝜆<1, and Ψ(𝜆,𝛾,𝑘,2) is given by (25). Clearly 𝐹∈ℳ_𝑞(𝜆,𝛾,𝑘). For this function (47) becomes Ψ𝜆,𝛾,𝑘,22𝐹(𝑧)≺𝑧1−𝛾+Ψ𝜆,𝛾,𝑘,2.1−𝑧 (56) It is easily verified that ΨminRe𝜆,𝛾,𝑘,22𝐹(𝑧)11−𝛾+Ψ𝜆,𝛾,𝑘,2=−2,𝑧∈𝑈. (57) This shows that the constant Ψ(𝜆,𝛾,𝑘,2)/2[1−𝛾+Ψ(𝜆,𝛾,𝑘,2)] cannot be replaced by any larger one.

5. Partial Sums

For a function 𝑓∈𝐴 given by (1) Silverman [12] and Silvia [13] investigated the partial sums 𝑓₁ and 𝑓_𝑚 defined by 𝑓₁(𝑧)=𝑧,𝑓_𝑚(𝑧)=𝑧+^𝑚_𝑛=2𝑎_𝑛𝑧^𝑛,(𝑚=2,3,…). (58) We consider in this section partial sums of functions in the class ℳ_𝑞(𝜆,𝛾,𝑘) and obtain sharp lower bounds for the ratios of real part of 𝑓 to 𝑓_𝑚(𝑧) and 𝑓^′ to 𝑓^′_𝑚.

Theorem 20. Let a function 𝑓 of the form (1) belong to the class ℳ_𝑞(𝜆,𝛾,𝑘) and satisfy the condition (17). Then 𝑓(𝑧)Re𝑓_𝑚(𝑧)1≥1−𝒞_𝑚+1,𝑧∈𝑈,𝑚∈𝑁,𝑓Re^𝑚(𝑧)𝑓(𝑧)≥𝒞_𝑚+11+𝒞_𝑚+1,𝑧∈𝑈,𝑚∈𝑁, (59)(60) where 𝒞_𝑛𝑛)∶=(1+𝑛𝜆−𝜆)(1+𝑘−𝑛𝛾+𝑘Φ1−𝛾^𝑞𝑛,𝜇. (61)

Proof. By (61) it is not difficult to verify that 𝒞_𝑛+1>𝒞_𝑛>1,𝑛=2,3,…. (62) Thus by Lemma 8 we have ^𝑚_𝑛=2𝑎_𝑛+𝒞^∞_{𝑚+1𝑛=𝑚+1}𝑎_𝑛≤^∞_𝑛=2𝒞_𝑛𝑎_𝑛≤1. (63) Setting 𝑔(𝑧)=𝒞_𝑚+1𝑓(𝑧)𝑓_𝑚(𝑧)−11−𝒞_𝑚+1𝒞=1+^𝑚+1∑^∞_𝑛=𝑚+1𝑎_𝑛𝑧^𝑛−11+∑^𝑚_𝑛=2𝑎_𝑛𝑧^𝑛−1, (64) it suffices to show that ℜ𝑔(𝑧)≥0,𝑧∈𝑈. (65) Applying (63), we find that 𝑔(𝑧)−1𝑔(𝑧)≤𝒞+1^𝑚+1∑^∞_𝑛=𝑚+1𝑎_𝑛2−2∑^𝑚_𝑛=2𝑎_𝑛−𝒞_𝑚+1∑^∞_𝑛=𝑚+1𝑎_𝑛≤1,𝑧∈𝑈, (66) which readily yields the assertion (59) of Theorem 20. In order to see that 𝑓(𝑧)𝑧=𝑧+^𝑚+1𝒞_𝑚+1,𝑧∈𝑈, (67) gives the sharp result, we observe that for 𝑧=𝑟𝑒^{𝑖𝜋/𝑚} we have 𝑓(𝑧)𝑓_𝑚(𝑧)𝑧=1+^𝑚𝒞_𝑚+1^𝑧→1−1−−−−−→1−𝒞_𝑚+1. (68) Similarly, if we take ℎ(𝑧)=1+𝒞_𝑚+1𝑓_𝑚(𝑧)𝑓(𝑧)−𝒞_𝑚+11+𝒞_𝑚+1=1−1+𝒞_𝑛+1∑^∞_𝑛=𝑚+1𝑎_𝑛𝑧^𝑛−11+∑^∞_𝑛=2𝑎_𝑛𝑧^𝑛−1,𝑧∈𝑈, (69) and making use of (63), we can deduce that ℎ(𝑧)−1ℎ(𝑧)≤+11+𝒞_𝑚+1∑^∞_𝑛=𝑚+1𝑎_𝑛2−2∑^𝑚_𝑛=2𝑎_𝑛−1−𝒞_𝑚+1∑^∞_𝑛=𝑚+1𝑎_𝑛≤1,𝑧∈𝑈, (70) which leads us immediately to the assertion (60) of Theorem 20. The bound in (60) is sharp for each 𝑚∈𝑁 with the extremal function 𝑓 given by (67), and the proof is complete.

Theorem 21. Let a function 𝑓 of the form (1) belong to the class ℳ_𝑞(𝜆,𝛾,𝑘) and satisfy the condition (17). Then 𝑓Re^′(𝑧)𝑓^′_𝑚(𝑧)≥1−𝑚+1𝒞_𝑚+1,𝑓Re^′𝑚(𝑧)𝑓^′(𝑧)≥𝒞_𝑚+1𝑚+1+𝒞_𝑚+1, (71) where 𝒞_𝑚 is defined by (61)

Proof. By setting 𝑔(𝑧)=𝒞_𝑚+1𝑓^′(𝑧)𝑓^′_𝑚(𝑧)−1−𝑚+1𝒞_𝑚+1ℎ(𝑧)=),𝑧∈𝑈,(𝑚+1+𝒞_𝑚+1𝑓^′_𝑚(𝑧)𝑓^′(𝑧)−𝒞_𝑚+1𝑚+1+𝒞_𝑚+1,𝑧∈𝑈, (72) the proof is analogous to that of Theorem 20, and we omit the details.

6. Integral Means

In [1], Silverman found that the function 𝑓₂(𝑧)=𝑧−(𝑧²/2) is often extremal over the family 𝒯. He applied this function to resolve his integral means inequality, conjectured in [14] and settled in [15], that ₀^2𝜋𝑓𝑟𝑒^𝑖𝜃𝜂𝑑𝜃≤₀^2𝜋𝑓₂𝑟𝑒^𝑖𝜃𝜂𝑑𝜃, (73) for all 𝑓∈𝒯, 𝜂>0, and 0<𝑟<1. In [15], he also proved his conjecture for the subclasses 𝒯^∗(𝛾) the class of starlike functions and 𝐶(𝛾) the class of convex functions with negative coefficients.

We recall the following lemma to prove our result on integral means inequality.

Lemma 22 (see [11]). If the functions 𝑓 and 𝑔 are analytic in 𝑈 with 𝑔≺𝑓, then for 𝜂>0, and 0<𝑟<1, ₀^2𝜋𝑔𝑟𝑒^𝑖𝜃𝜂𝑑𝜃≤₀^2𝜋𝑓𝑟𝑒^𝑖𝜃𝜂𝑑𝜃. (74)

Applying Lemmas 22 and 8 and Theorem 12, we prove the Silverman's conjecture for the functions in the family ℳ_𝑞(𝜆,𝛾,𝑘) by using known procedures.

Theorem 23. Suppose that 𝑓∈ℳ_𝑞(𝜆,𝛾,𝑘), 𝜂>0, 0≤𝜆<1, 0≤𝛾<1, 𝑘≥0, and 𝑓₂(𝑧) is defined by 𝑓₂(𝑧)=𝑧−1−𝛾Ψ𝑧𝜆,𝛾,𝑘,2², (75) where Ψ(𝜆,𝛾,𝑘,2) is defined in (25). Then for 𝑧=𝑟𝑒^𝑖𝜃, 0<𝑟<1, we have ₀^2𝜋𝑓(𝑧)^𝜂𝑑𝜃≤₀^2𝜋𝑓₂(𝑧)^𝜂𝑑𝜃. (76)

Concluding Remarks. In fact, by suitably specializing the values of 𝜆 and 𝑘, the results presented in this paper would find further applications for the class of univalent starlike functions with negative coefficients stated in Examples 4 and 5 of Section 1.