A Few Inequalities Established by Using Fractional Calculus and Their Applications to Certain Multivalently Analytic Functions

Abstract

By making use of different techniques given in Miller and Mocanu (2000) (and also in Jack (1971)), some recent results consisting of certain multivalently analytic functions given both in Irmak (2005) and in Irmak (2010) are firstly restated and some of their applications are then pointed out.

1. Introduction, Definitions, and Notations

Let 𝒜(𝑝) denote the class of functions 𝑓(𝑧) of the form: 𝑓(𝑧)=𝑧^𝑝+𝑎_𝑝+1𝑧^𝑝+1+𝑎_𝑝+2𝑧^𝑝+2+⋯+𝑎_𝑝+𝑛𝑧^𝑝+𝑛𝑎+⋯_𝑝+𝑛∈ℂ;𝑛,𝑝∈ℕ={1,2,3,…}, (1) which are analytic and multivalent in the open unit disk 𝕌={𝑧∶𝑧∈ℂand|𝑧|<1}, where ℂ is the set of complex numbers.

For some useful implications of the main results, there is a need to recall certain well-known definitions relating to geometric function theory. As is known, a function 𝑓(𝑧) belonging to the general class 𝒜(𝑝) is said to be multivalently starlike function, multivalently convex function, and multivalently close-to-convex function (with respect to the origin (𝑧=0)), if it satisfies ℜ𝑒[𝑧𝑓^′(𝑧)/𝑓(𝑧)]>0, ℜ𝑒[1+𝑧𝑓^′′(𝑧)/𝑓^′(𝑧)]>0, and ℜ𝑒[𝑧^1−𝑝𝑓^′(𝑧)]>0 in the open unit disk 𝕌, respectively. For the details of the definitions above, one may look over the works in [1–3].

Here and also throughout this paper, the symbol 𝐷^𝜇_𝑧{⋅} denotes an operator of fractional calculus (i.e., that fractional derivative(s)), which is defined as follows (cf., e.g., [4] and see (also) [2, 5, 6]).

Definition 1. Let 𝑓(𝑧) be an analytic function in a simply connected region of the 𝑧-plane containing the origin. The fractional integral of order 𝜇(𝜇>0) is defined by 𝐷_𝑧^−𝜇𝑓(𝑧=1)Γ𝜇^𝑧₀𝑓(𝜉)(𝑧−𝜉)^𝜇−1𝑑𝜉, (2) and the fractional derivative of order 𝜇(0≤𝜇<1) is defined by 𝐷^𝜇_𝑧𝑓(𝑧=1)Γ𝑑1−𝜇𝑑𝑧^𝑧₀𝑓(𝜉)(𝑧−𝜉)^−𝜇𝑑𝜉, (3) where the multiplicity of (𝑧−𝜉)^𝜇−1 involved in (2) and that of (𝑧−𝜉)^−𝜇 in (2) are removed by requiring log(𝑧−𝜉) to be real when 𝑧−𝜉>0.

Definition 2. Under the hypotheses of Definition 1, the fractional derivative of order 𝑚+𝜇(𝑚∈ℕ₀=ℕ∪{0};0≤𝜇<1) is defined by 𝐷_𝑧^𝑚+𝜇𝑓(𝑧=𝑑)^𝑚𝑑𝑧^𝑚𝐷^𝜇_𝑧𝑓(𝑧). (4)
The aim of this investigation is first to restate some recent results relating to certain multivalently analytic functions and then to point a number of their applications out. For proofs of the main results, the well-known assertions in [7] and [8, p. 33–35] are used (see, also, for similar proofs, [9–11]). The main results include fractional calculus and the main purpose of using fractional calculus is also to extend the scope of the main results and to reveal certain complex inequalities which can be associated with (analytic and) geometric function theory (see, for their details, [1–4, 9]). For certain results determined by fractional calculus, for example, one may refer to the works in [5, 6, 12, 13]. For some results between certain complex inequalities and analytic functions, one can also check the papers, for instance, in the references in [2, 12, 13].

2. Main Results and Certain Consequences

The following assertions (Lemmas 3 and 4 below) will be required for proving the main results.

Lemma 3 (see [7]). Let 𝑤(𝑧) be an analytic function in the unit disk 𝕌 with 𝑤(0)=0 and let 0<𝑟<1. If |𝑤(𝑧)| attains at 𝑧₀ its maximum value on the circle |𝑧|=𝑟, then 𝑧₀𝑤^′(𝑧₀)=𝑐𝑤(𝑧₀)(𝑐≥1).

Lemma 4 (see [8]). Let Ω⊂ℂ and suppose that the function 𝜓∶ℂ²×𝕌→ℂ satisfies 𝜓(𝑀𝑒^𝑖𝜃,𝐾𝑒^𝑖𝜃;𝑧)∉Ω for all 𝜃∈ℝ and 𝑧∈𝕌 with 𝐾≥𝑀𝑛(𝑛∈ℕ). If the function 𝑝(𝑧) is in the class ℋ[𝑎,𝑛]≡{𝑝(𝑧)∈ℋ(𝕌)∶𝑝(𝑧)=𝑎+𝑎_𝑛𝑧^𝑛+⋯(𝑧∈𝕌)} and 𝜓(𝑝(𝑧),𝑧𝑝^′(𝑧);𝑧)∈Ω for all 𝑧∈𝕌, then |𝑝(𝑧)|<𝑀(𝑧∈𝕌).

By using Lemma 3, we first prove the following theorem.

Theorem 5. Let 𝑧∈𝕌,𝛿∈ℝ⧵{0}=∶ℝ^∗,𝜆∈ℝ^∗,𝑝∈ℕ, and 0≤𝜇<1, and let the function 𝑓(𝑧)∈𝒜(𝑝) satisfy any one of the following inequalities:𝑧ℜ𝑒𝑧𝐷_𝑧^1+𝜇𝑓(𝑧)/𝐷^𝜇_𝑧𝑓(𝑧)^′𝑧𝐷_𝑧^1+𝜇𝑓(𝑧)/𝐷^𝜇_𝑧𝑓(𝑧)^𝛿−11+𝑧𝐷_𝑧^1+𝜇𝑓(𝑧)/𝐷^𝜇_𝑧𝑓(𝑧)^𝛿−𝑝−𝜇^𝛿<1>12𝜆𝛿if𝜆⋅𝛿>02𝜆𝛿if𝜆⋅𝛿<0. (5)Then ℜ𝑒1+𝑧𝐷_𝑧^1+𝜇𝑓(𝑧)𝐷^𝜇_𝑧𝑓(𝑧)^𝛿−𝑝−𝜇^𝛿𝜆>0, (6) where the value of each one of the above complex terms and their certain applications is taken to be its principal value and ℝ denotes the set of real numbers.

Proof. First of all, Definition 1 readily provides us with the following fractional derivative formula for a power function: 𝐷^𝜇_𝑧𝑧^𝜅=Γ(𝜅+1)Γ𝑧𝜅−𝜇+1^𝜅−𝜇𝜅>𝜇−1;0≤𝜇<1. (7) With the help of (7), let us define the function 𝑤(𝑧) by 1+𝑧𝐷_𝑧^1+𝜇𝑓(𝑧)𝐷^𝜇_𝑧𝑓(𝑧)^𝛿−𝑝−𝜇^𝛿𝜆(𝑧)().=1+𝑤𝑧∈𝕌 (8) Clearly, the function 𝑤(𝑧) is analytic in 𝕌 and 𝑤(0)=0. Differentiating both sides of the above identity, and if we make use of (8) once again, we easily arrive at 𝒢(𝑧)∶=𝜆𝛿𝑧𝑧𝐷_𝑧^1+𝜇𝑓(𝑧)𝐷^𝜇_𝑧𝑓(𝑧)^′𝑧𝐷_𝑧^1+𝜇𝑓(𝑧)𝐷^𝜇_𝑧𝑓(𝑧)^𝛿−1×1+𝑧𝐷_𝑧^1+𝜇𝑓(𝑧)𝐷^𝜇_𝑧𝑓(𝑧)^𝛿−𝑝−𝜇^𝛿−1=𝑧𝑤^′(𝑧)(𝑧).1+𝑤 (9) Suppose now that there exists a point 𝑧₀∈𝕌 such that max_{|𝑧|≤|𝑧0|}|𝑤(𝑧)|=|𝑤(𝑧₀)|=1. Applying Jack’s Lemma, we can write 𝑧₀𝑤^′(𝑧₀)=𝑐𝑤(𝑧₀)(𝑐≥1). Setting 𝑧=𝑧₀ and then putting 𝑤(𝑧₀)=𝑒^𝑖𝜃 in (9), we obtain ℜ𝑒𝒢𝑧₀=ℜ𝑒𝑐𝑒^𝑖𝜃1+𝑒^𝑖𝜃𝑐=ℜ𝑒1+𝑒^𝑖𝜃1+𝑒^𝑖𝜃2=𝑐2≥12, (10) where 𝜃≠𝜋 and 𝑐≥1. Hence, (10) is a contradiction to the conditions in (5). Consequently, we conclude from definition (8) that 1+𝑧𝐷_𝑧^1+𝜇𝑓(𝑧)𝐷^𝜇_𝑧𝑓(𝑧)^𝛿−𝑝−𝜇^𝛿𝜆=|𝑤(𝑧−1)|<1,𝑧∈𝕌,𝛿∈ℝ^∗,𝜆∈ℝ^∗,𝑝∈ℕ,0≤𝜇<1, (11) which implies (6). Thus, the proof of the Theorem 5 is completed.

The second theorem (below) is given without proof since its proof is similar to the proof of the Theorem 5.

Theorem 6. Let 𝑧∈𝕌, 𝛿∈ℝ^∗, 𝜆∈ℝ^∗, 𝑝∈ℕ, and 0≤𝜇<1. If the function 𝑓(𝑧) in the class 𝒜(𝑝) satisfies any one of the following conditions:𝑧𝐷ℜ𝑒^𝜇𝑧𝑓(𝑧)/𝑧^{𝑝−𝜇′}𝐷^𝜇_𝑧𝑓(𝑧)/𝑧^{𝑝−𝜇𝛿−1}1+𝐷^𝜇_𝑧𝑓(𝑧)/𝑧^{𝑝−𝜇𝛿}−Γ(𝑝+1)/Γ(𝑝−𝜇+1)^𝛿<1>12𝜆𝛿if𝜆⋅𝛿>02𝜆𝛿if𝜆⋅𝛿<0, (12)then 𝐷ℜ𝑒1+^𝜇𝑧{𝑓(𝑧)}𝑧^{𝑝−𝜇𝛿}−Γ(𝑝+1)Γ(𝑝−𝜇+1)^𝛿𝜆>0, (13) where the value of each one of the above complex terms and their certain applications is taken to be its principal value.

Theorem 7. Let 𝑧∈𝕌, 𝛿∈ℝ^∗, 𝜆∈ℝ^∗, 𝑀≥1, 𝑝∈ℕ, and 0≤𝜇<1. If 𝑓(𝑧)∈𝒜(𝑝) satisfies the following condition: ℜ𝑒𝜆𝛿⋅𝑧𝑧𝐷_𝑧^1+𝜇𝑓(𝑧)𝐷^𝜇_𝑧𝑓(𝑧)^′𝑧𝐷_𝑧^1+𝜇𝑓(𝑧)𝐷^𝜇_𝑧𝑓(𝑧)^𝛿−1×1+𝑧𝐷_𝑧^1+𝜇𝑓(𝑧)𝐷^𝜇_𝑧𝑓(𝑧)^𝛿−𝑝−𝜇^𝛿−1<𝑀,1+𝑀 (14) then 1+𝑧𝐷_𝑧^1+𝜇𝑓(𝑧)𝐷^𝜇_𝑧𝑓(𝑧)^𝛿−𝑝−𝜇^𝛿𝜆−1<𝑀. (15)

Proof. Define 𝑝(𝑧) by 1+𝑧𝐷_𝑧^1+𝜇𝑓(𝑧)𝐷^𝜇_𝑧𝑓(𝑧)^𝛿−𝑝−𝜇^𝛿𝜆(𝑧)().−1=𝑝𝑧∈𝕌 (16) It is clear that the function 𝑝(𝑧) is in the class ℋ[0,𝑛] (with, of course, 𝑛=1) and also it gives us 𝑧𝑝^′(𝑧)(𝑧)𝑧1+𝑝=𝜆𝛿⋅𝑧𝐷_𝑧^1+𝜇𝑓(𝑧)𝐷^𝜇_𝑧𝑓(𝑧)𝑧𝐷_𝑧^1+𝜇𝑓(𝑧)𝐷^𝜇_𝑧𝑓(𝑧)^𝛿−1×1+𝑧𝐷_𝑧^1+𝜇𝑓(𝑧)𝐷^𝜇_𝑧𝑓(𝑧)^𝛿−𝑝−𝜇^𝛿−1=∶Δ𝜇,𝛿;𝑓(𝑧). (17)
Let 𝜓()=𝑠𝑟,𝑠;𝑧{𝑤}<𝑀1+𝑟,Ω=𝑤∈ℂ∶ℜ𝑒;1+𝑀 (18) we then receive that 𝜓𝑝(𝑧),𝑧𝑝^′(𝑧);𝑧=Δ𝜇,𝛿;𝑓(𝑧)∈Ω (19) for all 𝑧 in 𝕌. Further, for any 𝜃∈ℝ,𝐾>𝑛𝑀≥𝑀, and 𝑧∈𝕌, because of 𝑀≥1 and 𝑛∈ℕ, we then obtain that ℜ𝑒𝜓𝑀𝑒^𝑖𝜃,𝐾𝑒^𝑖𝜃;𝑧=ℜ𝑒𝐾𝑒^𝑖𝜃1+𝑀𝑒^𝑖𝜃≥𝑛𝑀≥𝑀1+𝑀;1+𝑀 (20) that is, 𝜓(𝑀𝑒^𝑖𝜃,𝐾𝑒^𝑖𝜃;𝑧)∉Ω. Therefore, because of the Lemma 4, the function 𝑝(𝑧) defined in (16) follows the inequality in (15). This completes the desired proof.

Since the proof of the following theorem is similar to the proof of the Theorem 7, the details are here omitted.

Theorem 8. Let 𝑧∈𝕌, 𝛿∈ℝ⧵{0}=∶ℝ^∗, 𝑀≥1, 𝜆∈ℝ^∗, 𝑝∈ℕ, and 0≤𝜇<1. If the function 𝑓(𝑧) in the class 𝒜(𝑝) satisfies the following inequality:𝑧𝐷ℜ𝑒𝜆𝛿⋅^𝜇𝑧𝑓(𝑧)𝑧^{𝑝−𝜇′}𝐷^𝜇_𝑧𝑓(𝑧)𝑧^{𝑝−𝜇𝛿−1}×𝐷1+^𝜇𝑧𝑓(𝑧)𝑧^{𝑝−𝜇𝛿}−Γ(𝑝+1)Γ(𝑝−𝜇+1)^𝛿−1<𝑀,1+𝑀 (21) then 𝐷1+^𝜇𝑧𝑓(𝑧)𝑧^{𝑝−𝜇𝛿}−Γ(𝑝+1)Γ(𝑝−𝜇+1)^𝛿𝜆−1<𝑀. (22)

It is clear that, by suitably choosing the parameters 𝛿,𝜆,𝜇,𝑀 and also 𝑝 in the all theorems, one may easily receive several interesting and/or important results for analytic and geometric function theory (see, [1, 3], and also the others in the references). Since it is not possible to list all of them, we want to reveal only the useful certain consequences belonging to Theorems 5–8. They are in the following forms.

Letting 𝛿−1=𝜆−1=𝜇=0 and 𝛿=𝜆=1,𝜇→1—in the Theorem 5, we first get Corollaries 9 and 10, respectively.

Corollary 9. If a function 𝑓(𝑧)∈𝒜(𝑝) satisfies the inequality: 𝑧𝑓ℜ𝑒^′(𝑧)+𝑧𝑓^′′(𝑧)/𝑓(𝑧)−𝑧𝑓^′(𝑧)/𝑓(𝑧)²1+𝑧𝑓^′(𝑧)(𝑧)<1/𝑓−𝑝2(𝑧∈𝕌), (23) then ℜ𝑒𝑧𝑓^′(𝑧)𝑓(𝑧)>𝑝−1(𝑧∈𝕌); (24) that is, 𝑓(𝑧) is multivalently starlike in 𝕌.

Corollary 10. If a function 𝑓(𝑧)∈𝒜(𝑝) satisfies the inequality: 𝑧𝑓ℜ𝑒^′′(𝑧)+𝑧𝑓^′′′(𝑧)/𝑓^′(𝑧)−𝑧𝑓^′′(𝑧)/𝑓^′(𝑧)²2+𝑧𝑓^′′(𝑧)/𝑓^′(𝑧)<1−𝑝2(𝑧∈𝕌), (25) then ℜ𝑒1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)>𝑝−1(𝑧∈𝕌); (26) that is, 𝑓(𝑧) is multivalently convex in 𝕌.

Setting 𝛿=𝜆=1 and 𝜇→1—in the Theorem 6, we second obtain Corollary 11 (below).

Corollary 11. If a function 𝑓(𝑧)∈𝒜(𝑝) satisfies the inequality: ℜ𝑒𝑧𝑓^′′(𝑧)−𝑝−1𝑓^′(𝑧)𝑓^′(𝑧)−𝑝−1𝑧^𝑝−1<12(𝑧∈𝕌), (27) then 𝑓ℜ𝑒^′(𝑧)𝑧^𝑝−1>𝑝−1(𝑧∈𝕌); (28) that is, 𝑓(𝑧) is multivalently close-to-convex in 𝕌.

Remark 12. The above three consequences may be comparable with the results given by the papers in [12, 13].

Remark 13. Theorems 7 and 8 are equal to Theorems 5 and 6, respectively, when one takes 𝑀=1 in Theorems 7 and 8.

Conflict of Interests

The author declares that there is no conflict of interests.

Acknowledgment

The work in this investigation was supported by The Research Project financed by TÜBİTAK (The Scientific and Technological Research Council of TURKEY) under project no. TBAG-U-105T056.