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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 14, Issue 1 (Jan 2016)

Issues

Simple sufficient conditions for starlikeness and convexity for meromorphic functions

Pranay Goswami
  • School of Liberal Studies, Ambedkar University Delhi, Delhi-110006, India
  • Email:
/ Teodor Bulboacă
  • Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 400084 Cluj-Napoca, Romania
  • Email:
/ Rubayyi T. Alqahtani
  • Corresponding author
  • Email:
Published Online: 2016-08-12 | DOI: https://doi.org/10.1515/math-2016-0050

Abstract

In this paper we investigate some extensions of sufficient conditions for meromorphic multivalent functions in the open unit disk to be meromorphic multivalent starlike and convex of order α. Our results unify and extend some starlikeness and convexity conditions for meromorphic multivalent functions obtained by Xu et al. [2], and some interesting special cases are given.

Keywords: Meromorphic functions; Multivalent functions; Meromorphic starlike functions; Meromorphic convex functions; Analytic branch

MSC 2010: 30C45

1 Introduction and definitions

Let Σp, n denote the class meormorphic multivalent functions of the form f(z)=zp+k=nakpzkp(p,nN:={1,2,3}),(1)

which are analytic in the punctured unit disk U:=u{0}, where U:={zC:z<1}.

Definition 1.1:

  1. A function f ∈ Σp, n is said to be meromorphic starlike functions of order α, if it satisfies the inequality Rezf(z)f(z)>α,zU,

    for some real α (0 ≤ α < p), and we denote this subclass by ΣSp,n(α).

  2. A function f ∈ Σp, n is said to be meromorphic convex functions of order α, if it satisfies the inequality Re1+zf(z)f(z)>α,zU,

    for some real α (0 ≤ α < p), and we denote this subclass by ΣKp, n(α).

Let An denote the class of analytic functions in U of the form f(z)=z+k=nak+1zk+1,zU(nN).

Definition 1.2:

  1. Let Sn denote the class of n-starlike functions in U, i.e. fAn and satisfies Rezf(z)f(z)>0,zU.

  2. Further, we denote by Kn the class of n-convex functions in U, i.e. fAn and satisfies Re1+zf(z)f(z)>0,zU.

In the recent papers of Goyal et al. [1] and Xu et al. [2], the authors obtained some sufficient conditions for multivalent and meromorphic starlikeness and convexity, respectively. In this paper we will derive some extensions of these sufficient conditions for starlikeness and convexity of order α for meromorphic multivalent functions.

2 Main results

In order to find some simple sufficient conditions for the starlikeness and convexity of order α for a function f ∈ Σp, n, we will recall the following lemma due to P. T. Mocanu (see also [3]):

Lemma 2.1: If fAn and satisfies the inequality f(z)1<n+1(n+1)2+1,zU,then fSn.

Remark that, for the special case n = 1, this result was previously obtained in [5, Theorem 3].

Theorem 2.2: If f ∈ Σp, n, with f(z) ≠ 0 for all zU, satisfies the inequality zpf(z)1αpzf(z)f(z)+α+pα<(n+1)(pα)(n+1)2+1,zU,(2)for some real values of α (0 ≤ α < p), then fΣSp,n(α). (The power is the principal one).

Theorem 2.3: For f ∈ Σp, n, with f(z) ≠ 0 for all zU, let’s define the function h by h(z)=zzpf(z)1αp,zU(0α<p).(5)If h satisfies the inequality h(z)n+1(n+1)2+1,zU,(6)then fΣSp,n(α). (The power is the principal one).

Next, we will give some sufficient conditions for a function f ∈ Σp, n to be a convex function of order α.

Theorem 2.4: If f ∈ Σp, n, with f′(z) ≠ 0 for all zU, satisfies the inequality zp+1f(z)p1αp1+zf(z)f(z)+α+pα<(n+1)(pα)(n+1)2+1,zU,(7)for some real values of α (0 ≤ α < p), then f ∈ ΣKp, n(α). (The power is the principal one).

Theorem 2.5: If f ∈ Σp, n, with f′(z) ≠ 0 for all zU, satisfies the inequality 1zzp+1f(z)p1αp1+zf(z)f(z)+p(n+1)(pα)2(n+1)2+1,zU,(10)for some real values of α (0 ≤ α < p), then f ∈ ΣKp, n(α). (The power is the principal one).

Remarks 2.6:

  1. If we put n = 1 in Theorem 2.2 and Theorem 2.3, we get the results established by Xu et al. [2].

  2. For the special case n = 1, Theorem 2.4 and Theorem 2.5 represent the results of Xu et al. [2].

For f ∈ Σp, n, with f(z) ≠ 0 for all zU, let’s define the function F by F(z)=0z[tpf(t)]γdt,zU(γC),

where the power is the principal one. Thus, F(z)=z+γn+1anpzn+1+An, and considering this integral operator we derive the next result:

Theorem 2.7: If f ∈ Σp, n, with f(z) ≠ 0 for all zU, satisfies the inequality γ[zpf(z)]γzzf(z)f(z)+pn+12(n+1)2+1,zU,(12)for γ1p, then fΣSp,np+1γ. (The power is the principal one).

3 Special cases

Let’s consider the function f defined by f(z)=1zp1λ+λsinzzαp,zU,(15)

where 0 ≤ α < p, the power is the principal one, and assuming that the parameter λ ∈ ℂ is chosen such that 1λ1sinzz,zU.(16)

Using MAPLE™ software, from Figure 1a we may see that maxz11sinzz<0.18,(17)

therefore (16) holds whenever λ509=5.555..., and consequently, if λ ∈ ℂ satisfies this inequality then f ∈ Σp,2.

Using again MAPLE™ software, from Figure 1b we have that maxz1sin2z2<0.28,(18)

and a simple computation leads to zpf(z)1αpzf(z)f(z)+α+pα=2(pα)λsin2z2<2(pα)λ0.28,zU.

Thus, according to Theorem 2.2 we obtain the following special case:

Example 3.1: If λ ∈ ℂ and λ751410=1.6941..., then f(z)=1zp1λ+λsinzzαpΣSp,2(α),for some real values of α (0 ≤ α < p), where the power is the principal one.

Remark 3.2: For the function f given by (15), the function h defined by (5) is of the form h(z)=(1λ)z+λsinz,zU.Therefore, h(z)=λsinz, and using MAPLE™ software, the Figure 2a yields that maxz1sinz<1.2.Now, according to Theorem 2.3 we obtain the following special case:If λ ∈ ℂ and λ5210=0.79057..., then f(z)=1zp1λ+λsinzαpΣSp,2(α),for some real values of α (0 ≤ α < p), where the power is the principal one.If we compare the result given by Example 3.1 with the above one, for this special choice of the function f the Example 3.1 gives a better result.

As we already proved, if λ509=5.555... then the relation (16) holds. Therefore, there exists a function f ∈ Σp,2 such that f(z)=pzp+11λ+λsinzzαp,zU,

where 0 ≤ α < p, and the power is the principal one, assuming that λ ∈ ℂ is chosen such that λ509=5.555...

A simple computation combined with (18) shows that zp+1f(z)p1ap1+zf(z)f(z)+α+pα=2(pα)λsin2z2<2(pα)λ0.28,zU,

and from Theorem 2.4 we obtain the following special case:

Example 3.3: If λ ∈ ℂ and λ751410=1.6941..., then f ∈ Σp,2 with f(z)=pzp+11λ+λsinzzap,zU,(19)for some real values of α (0 ≤ α < p), is in ΣKp, n(α). (The power is the principal one).

Remark 3.4: Using MAPLE™ software, from Figure 2b we have that maxz1zcoszsinzz2<0.38.(20)From a simple computation combined with (20) we get 1zzp+1f(z)p1αp1+zf(z)f(z)+p=(pα)λzcoszsinzz2<(pα)λ0.38,zU,and using Theorem 2.5 we obtain the next special case:If λ ∈ ℂ and λ751910=1.2483..., then f ∈ Σp,2 that satisfies (19) has, moreover, the property that f ∈ ΣKp, n(α), for some real values of α (0 ≤; α < p), where the power is the principal one.Comparing the result given by Example 3.3 with the above one, for this special choice of the function f the Example 3.3 gives a better result.

As we proved at the beginning of this section, the function f ∈ Σp,2, where f(z)=1zp1λ+λsinzz1γ,zU,(21)

with λ ∈ ℂ, λ509=5.555..., and the power is the principal one. Using the inequality (20), we deduce that γ[zpf(z)]γzzf(z)f(z)+p=λzcoszsinzz2<λ0.38,zU,

and from Theorem 2.7 we obtain the next special case:

Example 3.5: If λ ∈ ℂ and λ751910=1.2483..., then the function f given by (21) is in ΣSp,2p+1γ. for γ1p. (The power is the principal one).

Using MAPLE™ software, we could check that the next inequalities hold (see Figures 3a, 3b, 4a, and 4b): maxz11+z2ez1z<0.22,(22) maxz1ez1z<0.73,(23) maxz1ez1<1.73,(24) maxz1zezez+1z212<0.51.(25)

From (22) and (23), using Theorem 2.2 we may easily obtain the following special case:

Example 3.6: If λ ∈ ℂ and λ3007310=1.2996..., then f(z)=1zp1+λez1z1z2αpΣSp,2(α),for some real values of α (0 ≤ α < p), where the power is the principal one.

Remark 3.7: From (22) and (24), according to Theorem 2.3 we could similarly obtain the next special case:If λ ∈ ℂ and λ30017310=0.54837..., then f(z)=1zp1+λez1z1z2αpΣSp,2(α),for some real values of α (0 ≤ α < p), where the power is the principal one.Thus, for this special choice of the function f the Example 3.6 gives a better result.

From (22) and (24), using Theorem 2.4 we easily get the next special case:

Example 3.8: If λ ∈ ℂ and λ3007310=1.2996..., then f ∈ Σp,2 with f(z)=pzp+11+λez1z1z2αp,zU,(26)for some real values of α (0 ≤ α < p), is in ΣKp, n(α). (The power is the principal one).

Remark 3.9: From (22) and (25), according to Theorem 2.5 we could similarly obtain the next special case:If λ ∈ ℂ and λ1505110=0.93008..., then f ∈ Σp,2 that satisfies (26) has, moreover, the property that f ∈ ΣKp, n(α), for some real values of α (0 ≤ α < p), where the power is the principal one.Consequently, for this special choice of the function f the Example 3.8 gives a better result.

Finally, from the inequalities (22) and (25), using Theorem 2.7 we obtain the next special case:

Example 3.10: If λ ∈ ℂ and λ1505110=0.93008..., then f(z)=1zp1+λez1z1z21γ,ΣSp,2p+1γ,for γ1p. (The power is the principal one).

We will omit the detailed proofs of the last three examples, since these are similar with the previous ones.

References

  • [1]

    Goyal S.P., Bansal S.K., Goswami P, Extensions of sufficient conditions for starlikeness and convexity of order α for multivalent function, Appl. Math. Lett., 2012, 25(11), 1993-1998

  • [2]

    Xu Y., Frasin B.A., Liu J., Certain sufficient conditions for starlikeness and convexity of meromorphically multivalent functions, Acta Math. Sci. Ser. B Engl. Ed., 2013, 33(5), 1300-1304

  • [3]

    Mocanu P.T., Oros Gh., Sufficient condition for starlikeness of order α, Int. J. Math. Math. Sci., 2001, 28(9), 557-560

  • [4]

    Mocanu P.T., Some simple criteria for starlikeness and convexity, Lib. Math. (N.S.), 1993, 13, 27-40

  • [5]

    Singh V., Univalent functions with bounded derivative in the unit disc, Indian J. Pure Appl. Math., 1977, 8, 1370-1377

About the article

Rubayyi T. Alqahtani: Department of Mathematics and Statistics, College of Science, Al-Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O.Box 65892, Riyadh 11566, Saudi Arabia


Received: 2016-01-31

Accepted: 2016-06-10

Published Online: 2016-08-12

Published in Print: 2016-01-01


Citation Information: Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2016-0050. Export Citation

© 2016 Goswami et al., published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. (CC BY-NC-ND 3.0)

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