# Differential Subordinations of Arithmetic and Geometric Means of Some Functionals Related to a Sector

Authors: A. Lecko; M. Lecko
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

Some general theorems on differential subordinations of some functionals connected with arithmetic and geometric means related to a sector are proved. These results unify a number of well known results concerning inclusion relation between the classes of analytic functions built with using arithmetic and geometric means.


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## Body

## 1. Introduction

Forr>0 let𝔻r={z∈ℂ:|z|<r}.Let𝔻=𝔻1.Let the functionsfandFbe analytic in the unit disc 𝔻. A functionfis called subordinate toF,written f≺F, ifFis univalent in 𝔻,f(0)=F(0) and f(𝔻)⊂F(𝔻).LetDbe a domain in ℂ2 and ψ:ℂ2⊃D→ℂ be an analytic function, and let p be a function analytic in 𝔻 with (p(z),zp′(z))∈D,z∈𝔻 andhbe a function analytic and univalent in 𝔻. The functionpis said to satisfy the first-order differential subordination if (1.1)ψ(p(z),zp′(z))≺h(z),ψ(p(0),0)=h(0),z∈D.The general theory of the differential subordinations has been studied intensively by many authors. A survey of this theory can by found in the monograph by Miller and Mocanu [1].Forβ∈(0,2] let (1.2)hβ(z)=(1+z1-z)β,z∈D.
It is clear thathβmaps univalently 𝔻 onto the sector of the angleβπsymmetrical with respect to the real axis with the vertex at the origin.In this paper we are interested in the following problem referring to (1.1) to find the constant ck(n,γ,α,β) so that to the following relation is true: (1.3){p(z)[1+αzp′(z)pk(z)]γ≺hck(n,α,γ,β)(z),z∈D}⟹p≺hβ,
with suitable assumptions on functionpand constants n,α,γ,β.For selected parameters n,γ,α,β the theorems presented here reduce to the well-known theorems proved by various authors. Particularly, results of this type can be applied to examine inclusion relation between subclasses of analytic functions defined with using arithmetic or geometric means of some functionals, for example, the class ofα-convex functions or γ-starlike functions.The lemma below that slightly generalizes a lemma proved by Miller and Mocanu [2] will be required in our investigation.Lemma 1.1 (see [2]).
Letq:𝔻¯→ℂ be a function analytic and univalent on  𝔻, injective on  ∂𝔻 and  q(0)=1. Let
(1.4)p(z)=1+∑k=n∞ckzk,z∈D,
be analytic in 𝔻,p≢1. Suppose that there exists a point z0∈𝔻 such that p(z0)∈∂q(𝔻) and
(1.5)p({z∈C:|z|<|z0|})⊂q(D).
Ifξ0=q-1(p(z0))and q′(ξ0) exists, then there exists anm≥nfor which
(1.6)z0p′(z0)=mξ0q′(ξ0).


## 2. Main Results

In the first theorem which follows directly from Theorem2.2 [3] we prove thatck(n,α,γ,β)≥β.Let us start with the following definition.Definition 2.1 (see [3]).
Letγ∈[0,1]andΦbe a function analytic in domain D⊂ℂ. Byℋ(γ,Φ)will be denoted the class of functionspanalytic in 𝔻 with p(0)=1,p≢1 and p(𝔻)⊂D such that the function
(2.1)Q(z)=p(z)[1+zp′(z)p(z)Φ(p(z))]γ,Q(0)=1,z∈D,
is well defined in 𝔻.Theorem 2.2 (see [3]).
Letγ∈[0,1],h a convex function such that  0∈h(𝔻)¯,h(0)=1,Φ a function analytic in a domain  D⊂ℂ such that  h(𝔻)⊂D, and  ReΦ(h(z))>0 for  z∈𝔻. If p∈ℋ(γ,Φ)and
(2.2)p(z)[1+zp′(z)p(z)Φ(p(z))]γ≺h(z),z∈D,
then
(2.3)p≺h.Definition 2.3.
Letk,n∈ℕ,α≥0andγ∈[0,1].Byℋk(n,α,γ)will be denoted the class of functionspanalytic in𝔻of the form (1.4) such that the function
(2.4)Q(z)=p(z)[1+αzp′(z)pk(z)]γ,Q(0)=1,z∈D,
is well defined in𝔻.Remark 2.4.
(1) Setting(2.5)Φ(w)=Φk,α(w)=αwk-1,w∈C∖{0},
we see that
(2.6)Hk(1,α,γ)=H(γ,Φk,α).
(2) For eachk,n,α,γas in Definition 2.3 the classℋk(n,α,γ)is nonempty. To see this take
(2.7)p(z)=1+cnzn,z∈D,
for sufficiently smallcn∈ℂ.
(3) Clearly, forγ=0the classℋk(n,α,0)contains all analytic functionspof the form (1.4).
(4) Letk=γ=1.Then
(2.8)Q(z)=p(z)+αzp′(z),z∈D.
Therefore the classℋ1(n,α,1)contains all analytic functionspof the form (1.4).
(5) Letpbe analytic function in𝔻of the form (1.4). Suppose thatp(z0)=0for somez0∈𝔻.Then
(2.9)p(z)=(z-z0)mp1(z),z∈D,
wherem≥1andp1is analytic function in𝔻withp1(z0)≠0forz∈𝔻.Then we have
(2.10)zp′(z)pk(z)=zm(z-z0)m-1p1(z)+(z-z0)mp1′(z)[(z-z0)mp1(z)]k=mp1(z)+(z-z0)p1′(z)(z-z0)m(k-1)+1p1k(z).
Hence we see that fork=1andγ∈(0,1)or fork≥2andγ∈(0,1]the function
(2.11)Q(z)=p(z)[1+αzp′(z)pk(z)]γ,z∈D,
has a pole atz0.Therefore for suchkandγwe see that everyp∈ℋk(n,α,γ)is nonvanishing inD.Theorem 2.5.
Letk∈ℕ,α≥0,γ∈[0,1]and β∈[0,1]be such that (k-1)β≤1.If p∈ℋk(1,α,γ)and
(2.12)p(z)[1+αzp′(z)pk(z)]γ≺hβ(z),z∈D,
then
(2.13)p≺hβ.Proof.
The caseα=0is evident so we assume thatα>0.Fork∈Nandα>0letΦ=Φk,αbe defined by (2.5). Forβ∈[0,1]the functionhβis convex with0∈∂h(𝔻).Since(k-1)β≤1,we have
(2.14)ReΦ(hβ(z))=ReΦ((1+z1-z)β)=αRe{(1-z1+z)(k-1)β}>0,z∈D.
Applying Theorem 2.2 withhβinstead ofhwe get the assertion.Now we prove two theorems were we improve the result of Theorem2.5. The problem (1.3) will be divided into two cases:k=1andk>1.First we consider the casek=1.The theorem below was proved in [4]. To be self-contained we include its proof.Theorem 2.6.
Fixn∈ℕ,α≥0and γ∈[0,1].Let β∈(0,β1(n,α,γ)],where β=β1(n,α,γ)is the solution of the equation
(2.15)β+c1(n,α,γ,β)=4-γ,
with
(2.16)c1(n,α,γ,β)=β+2γπarctan(nαβ).
Ifp∈ℋ1(n,α,γ)and
(2.17)p(z)[1+αzp′(z)p(z)]γ≺hc1(n,α,γ,β)(z),z∈D,
then
(2.18)p≺hβ.Proof.
(1) Assume thatα>0 and γ∈(0,1] since the cases γ=0 or α=0 are evident.
Suppose, on the contrary, thatpis not subordinate to hβ. Then, by the minimum principle for harmonic mappings there exists r0∈(0,1) such that
(2.19)p(Dr0)⊂hβ(D),
and one of the following cases hold:
(2.20)max⁡{Argp(z):z∈Dr0}=max⁡{Argp(z):|z|=r0}=βπ2,
or
(2.21)min⁡{Argp(z):z∈Dr0}=min⁡{Argp(z):|z|=r0}=-βπ2,
or
(2.22)p(z0)=0,
for somez0∈∂𝔻r0.
(2) Assume that (2.20) holds. Then there existsz0∈∂𝔻r0such that
(2.23)Argp(z0)=βπ2.
Letξ0=hβ-1(p(z0)).Thus
(2.24)p(z0)=hβ(ξ0)=(1+ξ01-ξ0)β≠0.
Thereforeξ0≠±1and
(2.25)1+ξ01-ξ0=xi
forx>0,that is,
(2.26)ξ0=xi-1xi+1.
Sinceξ0≠±1,sohβ′(ξ0)exists. Hence and by Lemma 1.1 there exists anm≥nfor which
(2.27)z0p′(z0)=mξ0hβ′(ξ0).
(3) Consequently,(2.28)p(z0)[1+αz0p′(z0)p(z0)]γ=hβ(ξ0)[1+mαξ0hβ′(ξ0)hβ(ξ0)]β=(xi)β[1+mαβ(1+x2)2xi]γ.
In view of the fact thatx>0let us take
(2.29)arg{1+mαβ(1+x2)2xi}∈[0,π2).
Hence and from (2.28) we have
(2.30)arg{p(z0)[1+αz0p′(z0)p(z0)]γ}=arg{(xi)β[1+mαβ(1+x2)2xi]γ}=βπ2+γarg{1+mαβ(1+x2)2xi}=βπ2+γarctan(mαβ(1+x2)2x).
By the above and by the fact thatm≥nwe have
(2.31)arg{p(z0)[1+αz0p′(z0)p(z0)]γ}≥βπ2+γarctan(nαβ(1+x2)2x)≥βπ2+γarctan(nαβ)=c1(n,α,γ,β)π2.
On the other hand, (2.30) yields
(2.32)arg{p(z0)[1+αz0p′(z0)p(z0)]γ}=βπ2+γarctan(mαβ(1+x2)2xi)≤(β+γ)π2.
Finally, the above and (2.31) lead to
(2.33)c1(n,α,γ,β)π2≤arg{p(z0)[1+αz0p′(z0)p(z0)]γ}≤(β+γ)π2≤2π-c1(n,α,γ,β)π2,
for allβ∈(0,β1(n,α,γ)].
Thus we arrive at a contradiction with (2.17) sop≺hβ.
(4) When (2.21) holds, we see thatx<0in (2.25). Next we finish the proof by similar argumentations like in the above.
(5) Assume now that (2.22) holds. In view of Remark 2.4 this is possible only whenk=γ=1.(a)
Forβ<1 the boundary ∂hβ(𝔻) has the corner at 0 of the angle βπ<π. Since p(∂𝔻r0) is an analytic curve, in view of (2.19) the case p(z0)=0 does not hold for β<1.(b)
Let nowβ≥1.
Assume thatp′(z0)≠0.Sincez0p′(z0)is an outer normal to the curvep(∂𝔻r0)atp(z0), by (2.19) we see that
(2.34)(3-β)π2=32π-βπ2≤arg{z0p′(z0)}≤βπ2+π2=(β+1)π2.
Hence taking into account that
(2.35)3-β≤c1(n,α,1,β),β+1≤4-c1(n,α,1,β),p(z0)+αz0p′(z0)=αz0p′(z0),
we deduce that
(2.36)c1(n,α,1,β)π2≤arg{p(z0)+αz0p′(z0)}≤(β+γ)π2≤2π-c1(n,α,1,β)π2,
for allβ∈(0,β1(n,α,γ)].In this way we arrive at a contradiction with (2.17) sop≺hβ.
Ifp′(z0)=0,then
(2.37)p(z0)+αz0p′(z0)=0∉hβ(D),
and once again we contradict (2.17).Special Cases
(1) The casen=1,α=1was proved in [5].
(2) The caseγ=1was proved in [6].
Corollary 2.7 (see [6]).
Letn∈ℕ,α≥0.Let β∈(0,β1(n,α,1)],where β=β1(n,α,1)is the solution of the equation
(2.38)β+c1(n,α,1,β)=3,
with
(2.39)c1(n,α,1,β)=β+2πarctan(nαβ).
Ifpis analytic function in𝔻of the form  (1.4) and
(2.40)p(z)+αzp′(z)≺hc1(n,α,1,β)(z),z∈D,
then
(2.41)p≺hβ.(3)
The caseγ=1,α=1was remarked [7].(4)
The caseγ=1,α=1,n=1was proved in detail in [8].Now we consider the problem (1.3) fork≥2.Theorem 2.8.
Letk≥2,n∈ℕ,α≥0,γ∈[0,1]and let β∈(0,1/(k-1)].If p∈ℋk(n,α,γ)and
(2.42)p(z)[1+αzp′(z)pk(z)]γ≺hck(n,α,γ,β)(z),z∈D,
then
(2.43)p≺hβ,
where
(2.44)ck(n,α,γ,β)=β+2γπ×arctan[nαβcos⁡((k-1)βπ/2)(1+(k-1)β)(1+(k-1)β)/2(1-(k-1)β)(1-(k-1)β)/2+nαβsin((k-1)βπ/2)].Proof.
(1) We repeat argumentation from Parts 1 and 2 of the proof of Theorem2.6.
(2) We have(2.45)p(z0)[1+αz0p′(z0)pk(z0)]γ=hβ(ξ0)[1+mαξ0hβ′(ξ0)hβk(ξ0)]β=(xi)β[1+mαβ(1+x2)2xi(xi)(k-1)β]γ=(xi)β[1+mαβ(1+x2)2x1+(k-1)βi1-(k-1)β]γ.
Sincex>0and1-(k-1)β≥0, we can take
(2.46)arg{1+mαβ(1+x2)2x1+(k-1)βi1-(k-1)β}∈[0,π2).
Hence
(2.47)arg{p(z0)[1+αz0p′(z0)pk(z0)]γ}=arg{(xi)β[1+mαβ(1+x2)2x1+(k-1)βi1-(k-1)β]γ}=βπ2+γarg{1+mαβ(1+x2)2x1+(k-1)βsin((k-1)βπ2)+imαβ(1+x2)2x1+(k-1)βcos⁡((k-1)βπ2)}=βπ2+γarctan[(mαβ(1+x2)/2x1+(k-1)β)cos⁡((k-1)βπ/2)1+(mαβ(1+x2)/2x1+(k-1)β)sin((k-1)βπ/2)].
Thus, from (2.46) and by the fact thatm≥nwe obtain
(2.48)arg{p(z0)[1+αz0p′(z0)pk(z0)]γ}≥βπ2+γarctan[nαβa(x)cos⁡((k-1)βπ/2)1+nαβa(x)sin((k-1)βπ/2)],
where
(2.49)a(x)=1+x22x1+(k-1)β,x≠0.
We have
(2.50)a′(x)=(1-(k-1)β)x2-(1+(k-1)β)2x2+(k-1)β,x≠0.
(3) Assume now that(k-1)β<1.Observe that the functionaattains its minimum at the point
(2.51)x0=1+(k-1)β1-(k-1)β.
Moreover
(2.52)a(x0)=1(1+(k-1)β)(1+(k-1)β)/2(1-(k-1)β)(1-(k-1)β)/2.
Hence, and from (2.48), we have
(2.53)arg{p(z0)[1+αz0p′(z0)pk(z0)]γ}≥βπ2+γarctan{nαβa(x0)cos⁡((k-1)βπ/2)1+nαβa(x0)sin((k-1)βπ/2)}=βπ2+γarctan[nαβcos⁡((k-1)βπ/2)(1+(k-1)β)(1+(k-1)β)/2(1-(k-1)β)(1-(k-1)β)/2+nαβsin((k-1)βπ/2)]=ck(n,α,γ,β)π2.
On the other hand, using the fact that0≤(k-1)βπ/2<π/2,from (2.47) we obtain
(2.54)arg{p(z0)[1+αz0p′(z0)pk(z0)]γ}=βπ2+γarctan[(mαβ(1+x2)/2x1+(k-1)β)cos⁡((k-1)βπ/2)1+(mαβ(1+x2)/2x1+(k-1)β)sin((k-1)βπ/2)]≤(β+γ)π2.
Finally, the above and (2.53) yield
(2.55)ck(n,α,γ,β)π2≤arg{p(z0)[1+αz0p′(z0)pk(z0)]γ}≤(β+γ)π2≤2π-ck(n,α,γ,β)π2,
for allβ∈(0,1/(k-1)].
Thus we arrive at a contradiction with (2.17) sop≺hβ.
(4) For(k-1)β=1we haveck(n,α,γ,β)=β.
This ends the proof of the theorem for the casex>0.
(5) When (2.21) holds, we see thatx<0in (2.25). Next we finish the proof by similar argumentations like in the above.
(6) Sinceβ≤1/(k-1)<1,arguing as in Part 5(a) of the proof  of Theorem 2.6 we see that the case (2.22) does not hold.Special Cases
(1)
β=1/(k-1).
Thenck(n,α,γ,β)=β.
Corollary 2.9.
Letk≥2,n∈ℕ,α≥0and γ∈[0,1].If p∈ℋk(n,α,γ)and
(2.56)p(z)[1+αzp′(z)pk(z)]γ≺h1/(k-1)(z),z∈D,
then
(2.57)p≺h1/(k-1).
(2)k=2,β=1.
Corollary 2.10.
Letn∈ℕ,α≥0and  γ∈[0,1].If p∈ℋ2(n,α,γ)and
(2.58)Re{p(z)[1+αzp′(z)p2(z)]γ}>0,z∈D,
then
(2.59)Re{p(z)}>0,z∈D.
(3) The casen=1,α=1was proved in [9].
(4)γ=1.
Corollary 2.11.
Letk≥2,n∈ℕ,α≥0and β∈(0,1/(k-1)].If pis a function   analytic in 𝔻of the form  (1.4) nonvanishing in 𝔻and
(2.60)p(z)+αzp′(z)pk-1(z)≺hck(n,α,1,β)(z),z∈D,
then
(2.61)p≺hβ,
whereck(n,α,1,β)is given by  (2.44).
(5)γ=1,k=2.Corollary 2.12.
Letn∈ℕ,α≥0, and β∈(0,1].If pis a function analytic in 𝔻of the form  (1.4) nonvanishing in 𝔻and
(2.62)p(z)+αzp′(z)p(z)≺hc2(n,α,1,β)(z),z∈D,
where
(2.63)c2(n,α,1,β)=β+2πarctan[nαβcos⁡(βπ/2)(1+β)(1+β)/2(1-β)(1-β)/2+nαβsin(βπ/2)],
then
(2.64)p≺hβ.(6) The caseγ=1,k=2,n=1,α=1was proved in [10]. The same result was reproved in [11] and once again in [12].(7)γ=1,k=2,β=1.Corollary 2.13.
Letn∈ℕ,α≥0,and β∈(0,1].If pis an analytic function in 𝔻of the form   (1.4) nonvanishing in 𝔻and
(2.65)Re{p(z)+αzp′(z)p(z)}>0,z∈D,
then
(2.66)Re{p(z)}>0,z∈D.


## 3. Applications

All this type results can be applied in the theory of analytic functions. Some results concerning the inclusion relations between subclasses of analytic functions can be formulated.Let𝒜(n),n∈N,denote the class of functions of the form (3.1)f(z)=z+∑k=n+1∞akzk,
which is analytic in𝔻.For short, let𝒜=𝒜(1).Also let𝒮denote the class of all functions in𝒜which are univalent in𝔻.To use theorems and corollaries listed in the previous section we put instead of the functionpsome functionals over the class𝒜(n),such asp(z)=f(z)/z,p(z)=zf′(z)/f(z)or the others. In this way the inclusion relations between selected subclasses of analytic functions can be obtained.


### 3.1. Arithmetic Means

(I)   γ=1,k=1.
(i)p(z)=f(z)/z,z∈𝔻,f∈𝒜(n),n∈N.
Forn∈ℕ,α≥0,β∈(0,1]letℛn(α,β)denote class of functionsf∈𝒜(n)such that
(3.2)(1-α)f(z)z+αf′(z)≺hβ(z),z∈D,
or, equivalently,
(3.3)|arg{(1-α)f(z)z+αf′(z)}|<βπ2,z∈D.
Using Corollary2.7 we have the following.
Corollary 3.1.
Letn∈ℕ,α≥0, and β∈(0,β1(n,α,1)].If f∈𝒜(n)and
(3.4)|arg{(1-α)f(z)z+αf′(z)}|<c1(n,α,1,β)π2,
then
(3.5)|arg{f(z)z}|<βπ2,z∈D.
The above result we can write in the following form.Corollary 3.2.
(3.6)Rn(α,c1(n,α,1,β))⊂Rn(0,β).
(ii)p(z)=f′(z),z∈𝔻,f∈𝒜(n),n∈ℕ
Forn∈ℕ,α≥0,β∈(0,1]let𝒯n(α,β)denote class of functionsf∈𝒜(n)such that
(3.7)f′(z)+αzf′′(z)≺hβ(z),z∈D,
or, equivalently,
(3.8)|arg{f′(z)+αzf′′(z)}|<βπ2,z∈D.
Remark 3.3.
(1) The class𝒯1(1,1)was introduced in [13].
(2) The class𝒯1(α,1)coincides with the classH(α,1,-1)studied in [14].
Observe thatf∈𝒯n(α,β)if and only ifzf′∈ℛn(α,β).
Using Corollary2.7 we have the following.
Corollary 3.4.
Letn∈ℕ,α≥0, and β∈(0,β1(n,α,1)].If f∈𝒜(n)and
(3.9)|arg{f′(z)+αzf′′(z)}|<c1(n,α,1,β)π2,z∈D,
then
(3.10)|arg{f′(z)}|<βπ2,z∈D.
Hence we have the following.
Corollary 3.5.
(3.11)Tn(α,c1(n,α,1,β))⊂Tn(0,β).(II)γ=1,k=2.
(ii)p(z)=zf′(z)/f(z),z∈𝔻,f∈𝒜(n),n∈ℕ.
Forn∈ℕ,α≥0,β∈(0,1]letℳn(α,β)denote class of functionsf∈𝒜(n)such thatf(z)f′(z)≠0forz∈𝔻and
(3.12)(1-α)zf′(z)f(z)+α(1+zf′′(z)f′(z))≺hβ(z),z∈D,
or, equivalently,
(3.13)|arg{(1-α)zf′(z)f(z)+α(1+zf′′(z)f′(z))}|<βπ2,z∈D.Remark 3.6.
(1) The classℳ1(α,1), that is, the class of so-called α-convex functions was introduced by Mocanu [15].
(2) The classℳ1(0,1) is identical with the class 𝒮* of starlike functions. The classℳ1(1,1)is identical with the class𝒞of convex functions.
(3) The classℳ1(0,β) denoted by𝒮*(β)were defined by Brannan and Kirwan [16] and, independently, by Stankiewicz [17, 18]. Functions in this class are called strongly starlike of orderβ.
The classℳ1(1,β) denoted by 𝒞*(β) contains functions called strongly convex of orderβ.
Using Corollary2.12 we have the following result proved by Marjono and Thomas [19].
Corollary 3.7.
Letn∈ℕ,α≥0, and β∈(0,1].If f∈𝒜(n)and
(3.14)|arg{(1-α)zf′(z)f(z)+α(1+zf′′(z)f′(z))}|<c2(n,α,1,β)π2,z∈D,
then
(3.15)|arg{zf′(z)f(z)}|<βπ2,z∈D.
Forα=1one has the result due to Nunokawa and Thomas [12]:Corollary 3.8.
Letn∈ℕand β∈(0,1].If f∈𝒜(n)and
(3.16)|arg{1+zf′′(z)f′(z)}|<c2(n,1,1,β)π2,z∈D,
then
(3.17)|arg{zf′(z)f(z)}|<β,z∈D.Corollary 3.9.
(3.18)Mn(α,c2(n,α,1,β))⊂Mn(0,β),M1(α,1)⊂S*,C⊂S*.


### 3.2. Geometric Mean

(I)   α=1,k=1.
(i)p(z)=f(z)/z,z∈𝔻,f∈𝒜(n),n∈ℕ.
Forn∈ℕ,β∈(0,1], andγ∈[0,1]letℒn(γ,β)denote class of functionsf∈𝒜(n)such that
(3.19)(f(z)z)1-γ(f′(z))γ≺hβ(z),z∈D,
or equivalently
(3.20)|arg{(f(z)z)1-γ(f′(z))γ}|<βπ2,z∈D.
Remark 3.10.
The classℒ1(γ,1)was introduced in [20].
Applying Theorem2.6 withα=1we have the following.
Corollary 3.11.
Letn∈ℕ,γ∈[0,1], andβ∈(0,β1(n,1,γ)].Iff∈𝒜(1)  and
(3.21)|arg{(f(z)z)1-γ(f′(z))γ}|<c1(n,1,γ,β)π2,z∈D,
then
(3.22)|arg{f(z)z}|<βπ2,z∈D.Corollary 3.12.
(3.23)Ln(γ,c1(n,1,γ,β))⊂Ln(0,β).(II)   α=1,k=2.
(ii)p(z)=zf′(z)/f(z),z∈𝔻,f∈𝒜(n),n∈ℕ.
Forn∈ℕ,β∈(0,1], andγ∈[0,1]let𝒮n*(γ,β)denote class of functionsf∈𝒜(n)such that
(3.24)(zf′(z)f(z))1-γ(1+zf′′(z)f′(z))γ≺hβ(z),z∈D,
or, equivalently,
(3.25)|arg{(zf′(z)f(z))1-γ(1+zf′′(z)f′(z))γ}|<βπ2,z∈D.
Remark 3.13.
(1) The class𝒮n*(γ,1),that is, the class of so-called γ-starlike functions was introduced by Lewandowski et al. [21].
(2) Clearly,(3.26)S1*(0,1)=S*,S1*(1,1)=C,S1*(0,β)=M1(0,β)=S*(β),S1*(1,β)=M1(1,β)=C*(β).
Using Theorem2.8 we obtain results due to Darus and Thomas [22].
Theorem 3.14.
Letn∈ℕ,γ∈[0,1], and β∈(0,1].If f∈𝒜(n)and
(3.27)|arg{(zf′(z)f(z))1-γ(1+zf′′(z)f′(z))γ}|<c2(n,1,γ,β)π2,z∈D,
then
(3.28)|arg{zf′(z)f(z)}|<βπ2,z∈D.Corollary 3.15.
(3.29)Sn*(γ;c2(n,1,γ,β))⊂Sn*(β),S1*(γ,1)⊂S*.
As further applications of Theorems2.6 and 2.8 we can use arbitrary well-defined functionals over the class𝒜(n).We recall two examples:
(1)(3.30)p(z)=Kaδ+1f(z)z,z∈D,f∈A(n),n∈N,a>0,δ≥0,
where the integral operatorKaδover the class𝒜(n)was defined by Komatu [23] as follows:
(3.31)Kaδf(z)=aδΓ(δ)∫01ta-2(log⁡1t)δ-1f(zt)dt,z∈D,
whereΓis the Gamma function;
(2)(3.32)p(z)=Lλf(z)z,z∈D,f∈A,λ≥-1,
where the operatorLλover the class𝒜called Ruscheweyh derivative [24] was defined as follows:
(3.33)Lλf(z)=z(1-z)λ+1*f(z),z∈D.


## 3.1. Arithmetic Means

(I)   γ=1,k=1.
(i)p(z)=f(z)/z,z∈𝔻,f∈𝒜(n),n∈N.
Forn∈ℕ,α≥0,β∈(0,1]letℛn(α,β)denote class of functionsf∈𝒜(n)such that
(3.2)(1-α)f(z)z+αf′(z)≺hβ(z),z∈D,
or, equivalently,
(3.3)|arg{(1-α)f(z)z+αf′(z)}|<βπ2,z∈D.
Using Corollary2.7 we have the following.
Corollary 3.1.
Letn∈ℕ,α≥0, and β∈(0,β1(n,α,1)].If f∈𝒜(n)and
(3.4)|arg{(1-α)f(z)z+αf′(z)}|<c1(n,α,1,β)π2,
then
(3.5)|arg{f(z)z}|<βπ2,z∈D.
The above result we can write in the following form.Corollary 3.2.
(3.6)Rn(α,c1(n,α,1,β))⊂Rn(0,β).
(ii)p(z)=f′(z),z∈𝔻,f∈𝒜(n),n∈ℕ
Forn∈ℕ,α≥0,β∈(0,1]let𝒯n(α,β)denote class of functionsf∈𝒜(n)such that
(3.7)f′(z)+αzf′′(z)≺hβ(z),z∈D,
or, equivalently,
(3.8)|arg{f′(z)+αzf′′(z)}|<βπ2,z∈D.
Remark 3.3.
(1) The class𝒯1(1,1)was introduced in [13].
(2) The class𝒯1(α,1)coincides with the classH(α,1,-1)studied in [14].
Observe thatf∈𝒯n(α,β)if and only ifzf′∈ℛn(α,β).
Using Corollary2.7 we have the following.
Corollary 3.4.
Letn∈ℕ,α≥0, and β∈(0,β1(n,α,1)].If f∈𝒜(n)and
(3.9)|arg{f′(z)+αzf′′(z)}|<c1(n,α,1,β)π2,z∈D,
then
(3.10)|arg{f′(z)}|<βπ2,z∈D.
Hence we have the following.
Corollary 3.5.
(3.11)Tn(α,c1(n,α,1,β))⊂Tn(0,β).(II)γ=1,k=2.
(ii)p(z)=zf′(z)/f(z),z∈𝔻,f∈𝒜(n),n∈ℕ.
Forn∈ℕ,α≥0,β∈(0,1]letℳn(α,β)denote class of functionsf∈𝒜(n)such thatf(z)f′(z)≠0forz∈𝔻and
(3.12)(1-α)zf′(z)f(z)+α(1+zf′′(z)f′(z))≺hβ(z),z∈D,
or, equivalently,
(3.13)|arg{(1-α)zf′(z)f(z)+α(1+zf′′(z)f′(z))}|<βπ2,z∈D.Remark 3.6.
(1) The classℳ1(α,1), that is, the class of so-called α-convex functions was introduced by Mocanu [15].
(2) The classℳ1(0,1) is identical with the class 𝒮* of starlike functions. The classℳ1(1,1)is identical with the class𝒞of convex functions.
(3) The classℳ1(0,β) denoted by𝒮*(β)were defined by Brannan and Kirwan [16] and, independently, by Stankiewicz [17, 18]. Functions in this class are called strongly starlike of orderβ.
The classℳ1(1,β) denoted by 𝒞*(β) contains functions called strongly convex of orderβ.
Using Corollary2.12 we have the following result proved by Marjono and Thomas [19].
Corollary 3.7.
Letn∈ℕ,α≥0, and β∈(0,1].If f∈𝒜(n)and
(3.14)|arg{(1-α)zf′(z)f(z)+α(1+zf′′(z)f′(z))}|<c2(n,α,1,β)π2,z∈D,
then
(3.15)|arg{zf′(z)f(z)}|<βπ2,z∈D.
Forα=1one has the result due to Nunokawa and Thomas [12]:Corollary 3.8.
Letn∈ℕand β∈(0,1].If f∈𝒜(n)and
(3.16)|arg{1+zf′′(z)f′(z)}|<c2(n,1,1,β)π2,z∈D,
then
(3.17)|arg{zf′(z)f(z)}|<β,z∈D.Corollary 3.9.
(3.18)Mn(α,c2(n,α,1,β))⊂Mn(0,β),M1(α,1)⊂S*,C⊂S*.


## 3.2. Geometric Mean

(I)   α=1,k=1.
(i)p(z)=f(z)/z,z∈𝔻,f∈𝒜(n),n∈ℕ.
Forn∈ℕ,β∈(0,1], andγ∈[0,1]letℒn(γ,β)denote class of functionsf∈𝒜(n)such that
(3.19)(f(z)z)1-γ(f′(z))γ≺hβ(z),z∈D,
or equivalently
(3.20)|arg{(f(z)z)1-γ(f′(z))γ}|<βπ2,z∈D.
Remark 3.10.
The classℒ1(γ,1)was introduced in [20].
Applying Theorem2.6 withα=1we have the following.
Corollary 3.11.
Letn∈ℕ,γ∈[0,1], andβ∈(0,β1(n,1,γ)].Iff∈𝒜(1)  and
(3.21)|arg{(f(z)z)1-γ(f′(z))γ}|<c1(n,1,γ,β)π2,z∈D,
then
(3.22)|arg{f(z)z}|<βπ2,z∈D.Corollary 3.12.
(3.23)Ln(γ,c1(n,1,γ,β))⊂Ln(0,β).(II)   α=1,k=2.
(ii)p(z)=zf′(z)/f(z),z∈𝔻,f∈𝒜(n),n∈ℕ.
Forn∈ℕ,β∈(0,1], andγ∈[0,1]let𝒮n*(γ,β)denote class of functionsf∈𝒜(n)such that
(3.24)(zf′(z)f(z))1-γ(1+zf′′(z)f′(z))γ≺hβ(z),z∈D,
or, equivalently,
(3.25)|arg{(zf′(z)f(z))1-γ(1+zf′′(z)f′(z))γ}|<βπ2,z∈D.
Remark 3.13.
(1) The class𝒮n*(γ,1),that is, the class of so-called γ-starlike functions was introduced by Lewandowski et al. [21].
(2) Clearly,(3.26)S1*(0,1)=S*,S1*(1,1)=C,S1*(0,β)=M1(0,β)=S*(β),S1*(1,β)=M1(1,β)=C*(β).
Using Theorem2.8 we obtain results due to Darus and Thomas [22].
Theorem 3.14.
Letn∈ℕ,γ∈[0,1], and β∈(0,1].If f∈𝒜(n)and
(3.27)|arg{(zf′(z)f(z))1-γ(1+zf′′(z)f′(z))γ}|<c2(n,1,γ,β)π2,z∈D,
then
(3.28)|arg{zf′(z)f(z)}|<βπ2,z∈D.Corollary 3.15.
(3.29)Sn*(γ;c2(n,1,γ,β))⊂Sn*(β),S1*(γ,1)⊂S*.
As further applications of Theorems2.6 and 2.8 we can use arbitrary well-defined functionals over the class𝒜(n).We recall two examples:
(1)(3.30)p(z)=Kaδ+1f(z)z,z∈D,f∈A(n),n∈N,a>0,δ≥0,
where the integral operatorKaδover the class𝒜(n)was defined by Komatu [23] as follows:
(3.31)Kaδf(z)=aδΓ(δ)∫01ta-2(log⁡1t)δ-1f(zt)dt,z∈D,
whereΓis the Gamma function;
(2)(3.32)p(z)=Lλf(z)z,z∈D,f∈A,λ≥-1,
where the operatorLλover the class𝒜called Ruscheweyh derivative [24] was defined as follows:
(3.33)Lλf(z)=z(1-z)λ+1*f(z),z∈D.



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*Source: 205845-2011-05-26.xml*