We obtain several sufficient conditions for λ-spirallike and λ-Robertson functions of complex order by making use of the well-known Jack Lemma.

1. Introduction and Definitions

Let 𝒜 denote the class of functions of the form
(1)f(z)=z+∑n=2∞anzn,
which are analytic in the open unit disk 𝒰={z:z∈ℂ and |z|<1}. A function f(z)∈𝒜 is said to be the λ-spirallike function of complex order b and type α(0≤α<1) in 𝒰, denoted by 𝒮αλ(b) if and only if
(2)ℜe{1+eiλbcosλ(zf′(z)f(z)-1)}>α(z∈𝒰),
for some real numbers λ∈ℝ with |λ|<π/2 and b∈ℂ*:=ℂ∖{0}. Furthermore, a function f(z)∈𝒜 is also said to be the λ-Robertson function of complex order b and type α(0≤α<1) in 𝒰 if and only if
(3)ℜe{1+eiλbcosλ(zf′′(z)f′(z))}>α(z∈𝒰),
for some real numbers λ∈ℝ with |λ|<π/2 and b∈ℂ*. We denote this class by 𝒞αλ(b).

Noting that the above function classes include several subclasses which have important role in the analytic and geometric function theory. From this reason, we want to state some of them.

𝒮0λ(b)=:𝒮λ(b)(b∈ℂ*;|λ|<π/2) is the class of the λ-spirallike function of complex order b introduced and studied by Al-Oboudi and Haidan [1].

𝒞0λ(b)=:𝒞λ(b)(b∈ℂ*;|λ|<π/2) is the class of the λ-Robertson function of complex order b (see, [2]).

𝒮α0(b)=:𝒮α*(b)(b∈ℂ*;0≤α<1) is the class of starlike functions of complex order b and type α introduced and studied by the author [3], and 𝒮α*(1)=:𝒮α* is said to be the starlike functions class of order α(0≤α<1) and was studied by Robertson [4].

𝒞α0(b)=:𝒞α(b)(b∈ℂ*;0≤α<1) is the class of convex functions of complex order b and type α introduced and studied by the author [3], and 𝒞α(1)=:𝒞α is said to be the convex functions class of order α(0≤α<1) and was studied by Robertson [4].

𝒮0λ(1)=:𝒮λ(|λ|<π/2) is known the λ-spirallike univalent functions class and was defined by Spacek [5], 𝒮00(b)=:𝒮(b)(b∈ℂ*) is said to be the starlike functions class of complex order and was studied by Nasr and Aouf [6], and 𝒮0λ(1-β)=:𝒮λ(β)(0≤β<1) is known the λ-spirallike functions class of order β and was studied by Libera [7].

𝒞0λ(1)=:𝒞λ(|λ|<π/2) is known the λ-Robertson type functions class and was first studied by Robertson [8], 𝒞00(b)=:𝒞(b) is called the convex functions class of complex order and was studied by Wiatrowski [9], Nasr and Aouf [10] and Aouf [11], and 𝒞0λ(1-β)=:𝒞λ(β)(0≤β<1) is known the λ-Robertson type functions class of order β and was studied by Chichra [12].

In this paper, we obtain several sufficient conditions for the analytic functions f(z) belonging to the classes 𝒮αλ(b), 𝒞αλ(b), 𝒮α*(b), 𝒞α(b), 𝒮(b), 𝒞(b), 𝒮α*, and 𝒞α by making use of the well-known Jack Lemma [13].

2. Main Result

In order to derive our main result, we have to recall here the following Jack Lemma.

Lemma 1 (see [<xref ref-type="bibr" rid="B7">13</xref>]).

Let w(z) be analytic in 𝒰 such that w(0)=0. Then, if |w(z)| attains its maximum value on circle |z|=r<1 at a point z0∈𝒰, one has
(4)z0w′(z0)=kw(z0),
where k>1 is a real number.

Now, with the help of Lemma 1, we can prove the following result.

Theorem 2.

Let f(z)∈𝒜, ζ,b∈ℂ* and |λ|<π/2, and let F(z) be defined by
(5)F(z)=(1-μ)f(z)+μzf′(z),(0≤μ≤1).
If F(z) satisfies any of the following inequalities:
(6)|((z[zF′(z)]′/F(z))-(zF′(z)/F(z))2)(1+(eiλ/bcosλ)((zF′(z)/F(z))-1))((1+(eiλ/bcosλ)(zF′(z)/F(z)-1))ζ-1)|<|bζ|(cosλ2-α),(z∈𝒰;0≤α<1),(7)|((z[zF′(z)]′/F(z))-(zF′(z)/F(z))2)1+(eiλ/bcosλ)((zF′(z)/F(z))-1)|<|bζ|((1-α)cosλ2-α),(z∈𝒰;0≤α<1),(8)|((z[zF′(z)]′/F(z))-(zF′(z)/F(z))2)((1+(eiλ/bcosλ)((zF′(z)/F(z))-1))ζ+1)|<|bζ|((1-α)cosλ(2-α)2),(z∈𝒰;0≤α<1),(9)|(1+eiλbcosλ(zF′(z)F(z)-1))ζ-1×(z[zF′(z)]′F(z)-(zF′(z)F(z))2)|<|bζ|(1-α)cosλ,(z∈𝒰;0≤α<1),(10)Re{((1+(eiλ/bcosλ)((zF′(z)/F(z))-1))ζ-1)-1((1+(eiλ/bcosλ)((zF′(z)/F(z))-1))ζ-1×(z[zF′(z)]′/F(z))-(zF′(z)/F(z))2(1+(eiλ/bcosλ)(zF′(z)/F(z)-1))ζ-1)×((1+(eiλ/bcosλ)((zF′(z)/F(z))-1))ζ-1)-1}<cosλ|ζ|2Re{bζ¯e-iλ},(z∈𝒰),
then
(11)Re(1+eiλbcosλ(zF′(z)F(z)-1))ζ>α,(z∈𝒰;0≤α<1).
The powers are taken by their principal value.

Proof.

Define a function w(z) by
(12)(1+eiλbcosλ(zF′(z)F(z)-1))ζ=1+(1-α)w(z).

Then, w(z) is analytic in 𝒰 and w(0)=0. It follows from (12) that
(13)((z[zF′(z)]′/F(z))-(zF′(z)/F(z))2)1+(eiλ/bcosλ)((zF′(z)/F(z))-1)+(1+eiλbcosλ(zF′(z)F(z)-1))ζ-1=(1-α)w(z)[1+be-iλcosλζzw′(z)w(z)11+(1-α)w(z)].

Thus, we have
(14)𝔽1(z)=((z[zF′(z)]′/F(z))-(zF′(z)/F(z))2)(1+(eiλ/bcosλ)((zF′(z)/F(z))-1))((1+(eiλ/bcosλ)((zF′(z)/F(z))-1))ζ-1)=be-iλcosλζzw′(z)w(z)11+(1-α)w(z),(15)𝔽2(z)=(z[zF′(z)]′/F(z))-(zF′(z)/F(z))21+(eiλ/bcosλ)((zF′(z)/F(z))-1)=be-iλcosλζ((1-α)zw′(z)1+(1-α)w(z)),(16)𝔽3(z)=(z[zF′(z)]′/F(z))-(zF′(z)/F(z))2(1+(eiλ/bcosλ)((zF′(z)/F(z))-1))ζ+1=be-iλcosλζ((1-α)zw′(z)[1+(1-α)w(z)]2),(17)𝔽4(z)=(1+eiλbcosλ(zF′(z)F(z)-1))ζ-1×(z[zF′(z)]′F(z)-(zF′(z)F(z))2)=be-iλcosλζ(1-α)zw′(z),(18)𝔽5(z)=(1+(eiλ/bcosλ)((zF′(z)/F(z))-1))ζ-1×((z[zF′(z)]′/F(z))-(zF′(z)/F(z))2)(1+(eiλ/bcosλ)(zF′(z)/F(z)-1))ζ-1×((1+(eiλ/bcosλ)((zF′(z)/F(z))-1))ζ-1)-1=be-iλcosλζzw′(z)w(z).

We claim that |w(z)|<1 in 𝒰. For otherwise, by Lemma 1, there exists z0∈𝒰 such that z0w′(z0)=kw(z0), where |w(z0)|=1 and k>1. Therefore, (14)–(18) yield
(19)|𝔽1(z0)|=|be-iλcosλζz0w′(z0)w(z0)11+(1-α)w(z0)|=(cosλ)|bζ|k|w(z0)||1+(1-α)w(z0)|≥|bζ|(cosλ2-α),|𝔽2(z0)|=|be-iλcosλζ((1-α)z0w′(z0)1+(1-α)w(z0))|=(cosλ)|bζ|k(1-α)|w(z0)||1+(1-α)w(z0)|≥|bζ|((1-α)cosλ2-α),|𝔽3(z0)|=|be-iλcosλζ((1-α)z0w′(z0)[1+(1-α)w(z0)]2)|=(cosλ)|bζ|k(1-α)|w(z0)|[|1+(1-α)w(z0)|]2≥|bζ|((1-α)cosλ(2-α)2),|𝔽4(z0)|=|be-iλcosλζ(1-α)z0w′(z0)|=cosλ|bζ|k(1-α)|w(z0)|≥|bζ|(1-α)cosλ,Re{𝔽5(z0)}=Re{be-iλcosλζz0w′(z0)w(z0)}=k(cosλ)Re{bζe-iλ}≥cosλ|ζ|2Re{bζ¯e-iλ},
which contradicts our assumptions (6)–(10), respectively. Therefore, |w(z)|<1 holds true for all z∈𝒰. We finally have
(20)|(1+eiλbcosλ(zF′(z)F(z)-1))ζ-1|<1-α,
thus, we have
(21)Re(1+eiλbcosλ(zF′(z)F(z)-1))ζ>α,(z∈𝒰;0≤α<1).

Remark 3.

Taking different choices of α,λ, μ, and ζ in Theorem 2, we obtain new sufficient conditions for functions to be in the classes 𝒮αλ(b), 𝒞αλ(b),𝒮α*(b), 𝒞α(b), 𝒮(b), 𝒞(b), 𝒮α*, and 𝒞α.

Acknowledgment

The author would like to thank the referee for his helpful comments and suggestions.

Al-OboudiF. M.HaidanM. M.Spirallike functions of complex orderAoufM. K.Al-OboudiF. M.HaidanM. M.On some results for λ-Roberston functions ofcomplex orderFrasinB. A.Family of analytic functions of complex orderRobertsonM. I. S.On the theory of univalent functionsSpacekL.Contribution a la theori des function univalentesNasrM. A.AoufM. K.Starlike function of complex orderLiberaR. J.Univalent α-spiral functionsRobertsonM. S.Univalent functions f(z) for which zf'(z) is spirallikeWiatrowskiP.The coefficients of a certain family of holomorphic functionsNasrM. A.AoufM. K.On convex functions of complex orderAoufM. K.p-valent classes related to convex functions of complex orderChichraP. N.Regular functions f(z) for which zf'(z) is α-spiral-likeJackI. S.Functions starlike and convex of order α