Let D denote the open unit disk and let S denote the class of normalized univalent functions which are analytic in D. Let Co(α) be the class of concave functions f∈S, which have the condition that the opening angle of f(D) at infinity is less than or equal to πα, α∈(1,2]. In this paper, we find a sufficient condition for the Gaussian hypergeometric functions to be in the class Co(α). And we define a class Co(α,A,B), (-1≤B<A≤1), which is a subclass of Co(α) and we find the set of variabilities for the functional (1-|z|2)(f″(z)/f′(z)) for f∈Co(α,A,B). This gives sharp upper and lower estimates for the pre-Schwarzian norm of functions in Co(α,A,B). We also give a characterization for functions in Co(α,A,B) in terms of Hadamard product.

1. Introduction

Let H denote the class of functions analytic in the unit disk D={z∈C:|z|<1}. We denote the class of locally univalent functions by LU. Let A denote the class of functions f∈H with normalization f(0)=f′(0)-1=0 and let S be the class of functions in A that are univalent in D. Also we define the subclass K⊂S of convex functions whenever f(D) is a convex domain.

A function f:D→C is said to belong to the family Co(α) if f satisfies the following conditions:

fis analytic in D with the standard normalization f(0)=f′(0)-1=0. In addition, it satisfies f(1)=∞.

f maps D conformally onto a set whose complement with respect to C is convex.

The opening angle of f(D) at ∞ is less than or equal to πα, α∈(1,2].

The class Co(α) is referred to as the class of concave univalent functions. We note that for f∈Co(α), α∈(1,2], the closed set C∖f(D) is convex and unbounded. We observe that Co(2) contains the classes Co(α), α∈(1,2].

Avkhadiev and Wirths [1] found the analytic characterization for functions in Co(α), α∈(1,2]: f∈Co(α) if and only if (1)Re2α-1α+121+z1-z-1-zf′′zf′z>0,z∈D.For f∈LU, the pre-Schwarzian derivative Tf is defined by Tf=f′′/f′ and we define the norm of Tf by (2)Tf=supz∈D1-z2Tfz.It is well known that Tf≤6 for f∈S and Tf≤4 for f∈K. In [2], Bhowmik et al. obtained the estimate of the pre-Schwarzian norm for functions f∈Co(α) as the following: (3)4≤Tf≤2α+2,f∈Co(α).For more investigation of concave functions, we may refer to [3–7].

We say that f is subordinate to F in D, written as f≺F, if and only if f(z)=F(w(z)) for some Schwarz functions w(z), w(0)=0, and |w(z)|<1, z∈D. If F(z) is univalent in D, then the subordination f≺F is equivalent to f(0)=F(0) and f(D)⊂F(D).

By using the subordination, we define a subclass of concave functions as follows.

Definition 1.

Let A and B be real numbers such that -1≤B<A≤1. The function f∈A belongs to the class Co(α,A,B) if f satisfies the following:(4)2α-1α+121+z1-z-1-zf′′zf′z≺1+Az1+Bzz∈D.

Note that Co(α,1,-1)≡Co(α).

Let a, b, and c be complex numbers with c≠0,-1,-2,…. We define the Gaussian hypergeometric function F12(a,b,c;z) by (5)F12(a,b,c;z)≔Fa,b,c;z=∑k=0∞akbkckzkk!,where (γ)k is Pochhammer symbol defined, in terms of Gamma function Γ, by (6)γk≔Γγ+kΓγγk=1,k=0γγ+1⋯γ+k-1k∈N.We note that the Gaussian hypergeometric function F satisfies the hypergeometric differential equation(7)z1-zF′′(z)+c-a+b+1zF′(z)-abF(z)=0.

The Gaussian hypergeometric function has been studied extensively by various authors [8–13]. In particular, univalency, close-to-convexity, starlikeness, convexity, and various other properties associated with these hypergeometric functions were investigated based on the conditions of a, b, and c in [14–17].

If f∈H and g∈H given by(8)fz=∑n=0∞anzn,g(z)=∑n=0∞bnzn,then Hadamard product (or convolution) f∗g∈H of f and g is defined (as usual) by (9)f∗gz=∑n=0∞anbnzn.

In this paper, we find a sufficient condition for the Gaussian hypergeometric functions to be in the class Co(α). And we find the set of variabilities for the functional (1-|z|2)Tf(z) and as a consequence of this we derive upper and lower bounds for the pre-Schwarzian norm Tf, for functions f in Co(α,A,B). And we give a representation formula in terms of Hadamard product for functions in Co(α,A,B).

2. A Sufficient Condition for Functions to Be in <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M113"><mml:mtext>Co</mml:mtext><mml:mo mathvariant="bold">(</mml:mo><mml:mi>α</mml:mi><mml:mo mathvariant="bold">)</mml:mo></mml:math></inline-formula>

In this section, we investigate a sufficient condition for the Gaussian hypergeometric functions to be in the class Co(α). The proof of our result in this section is based upon the following lemmas.

Lemma 2 (see [<xref ref-type="bibr" rid="B18">18</xref>], Miller and Mocanu, p. 35).

Let Ω be a set in the complex plane C and let b be a complex number such that Re(b)>0. Suppose that the function ψ:C2×D→C satisfies the condition (10)ψ(ix,y;z)∉Ωfor all real x, y≤-|b-ix|2/(2Re(b)) and all z∈D. If the function p(z) defined by pz=b+b1z+b2z2+⋯ is analytic in D and if (11)ψ(p(z),zp′(z);z)∈Ω,then Re{p(z)}>0 in D.

Lemma 3 (see [<xref ref-type="bibr" rid="B18">18</xref>], Miller and Mocanu, p. 239).

If a, b, and c are real and satisfy(12)-1≤b≤c,a∈-2,0∪c-1,c+1,then F′a,b,c;z≠0.

Theorem 4.

Let α∈(1,2] and a, b, and c be real and satisfy (12) and(13)α+c-1>-α+a+b-1,(14)ReDzBz¯≤0,in D, where(15)Bz=2α-11-z-α+a+b-1z-α+c-1,Dz=α+3α-2a-2b-1+4a+1b+1z2+2(α+3)(α+c)-2(α-1)(a+b+1)-4a+1b+1z+α-1α-1+2c.Then, (16)cabFa,b,c;z-1∈Co(α).

Proof.

Let (17)fz=cabFa,b,c;z-1.Then, f satisfies f(0)=0, f′(0)=1, and (18)F(z)≔F(a,b,c;z)=abcf(z)+1.From the hypergeometric differential equation (7), we have(19)z1-zf′′(z)+c-a+b+1zf′(z)-abf(z)-c=0.From condition (12) and Lemma 3, we have (20)f′z=cabF′(a,b,c;z)≠0in D. If we set(21)p(z)=2α-1α+121+z1-z-1-zf′′zf′z,then p is analytic in D and satisfies p(0)=1. Furthermore, we have(22)α-11-zf′zp(z)=α+11+zf′z-21-zf′(z)-2z1-zf′′z.Hence, we get(23)2z1-zf′′z=α-1+α+3z-α-11-zpzf′z.If we use this substitution in (19), then(24)α-1+α+3z-α-11-zpzf′z+2c-a+b+1zf′z-2abfz-2c=0.Differentiating (24), we have(25)α+3+α-1pz-α-11-zp′zf′z+α-1+α+3z-α-11-zpzf′′z-2a+b+1f′z+2c-a+b+1zf′′z-2abf′z=0.Substituting (23) in (25), we have(26)2α+3+α-1pz-α-11-zp′zz(1-z)+α-1+α+3z-α-11-zpz2-4(a+b+1)z(1-z)+2c-a+b+1z·α-1+α+3z-α-11-zpz-4abz(1-z)=0.And this equation leads us to the following first order differential equation:(27)Azp2z+Bzpz+Czzp′z+Dz=0,where (28)A(z)=α-121-z2,Bz=2(α-1)(1-z)-α+a+b-1z-α+c-1,Cz=-2α-11-z2,Dz={(α+3)(α-2a-2b-1)+4(a+1)(b+1)}z2+2(α+3)(α+c)-2(α-1)(a+b+1)-4(a+1)(b+1)z+(α-1)(α-1+2c).Since inequality (13) implies B(z)≠0 in D, we can rewrite (27) in the form (29)Jzp2+pz+Kzzp′z+Lz=0,where(30)Jz=AzBz,Kz=CzBz,L(z)=D(z)B(z).A bilinear transformation w(z)=(1+Az)/(1+Bz), with -1≤A≤1 and -1≤B≤1, has the property that Re{w(z)}>0, for z∈D. From the condition α∈(1,2] and (13), we have(31)Re{K(z)}=1α+c-1Re1-z1+((α-a-b+1)/(α+c-1))z>0in D and(32)ReJz+12Kz=2-α2α+c-1Re1-z1+α-a-b+1/α+c-1z≥0in D. Now, we let Ω={0} and define a function ψ:C2×D→C by (33)ψ(r,s;z)=J(z)r2+r+K(z)s+L(z).Then, (27) becomes (34)ψ(p(z),zp′(z);z)∈Ω.By using (14), (31), and (32), we obtain(35)Reψiρ,σ;z=-ReJzρ2+ReKzσ+ReLz≤-ReJz+12Kzρ2-12ReKz+ReLz<ReLz=ReDzBz¯Bz2≤0,for all z∈D and all real ρ, σ with σ≤-(1+ρ2)/2. By Lemma 2, we have Re{p(z)}>0 in D, which shows that f∈Co(α).

Example 5.

If we take α=2, a=3, b=1, and c=2, then we can easily check that conditions (12) and (13) are satisfied. Furthermore, (36)Reφ1z≤0(z∈D),where(37)φ1(z)=D(z)Bz¯=21-z-3+z¯5-2z-3z2(see Figure 1). Hence, (38)f1(z)=23F3,1,2;z=23-1+1-z/21-z2belongs to Co(2), as shown in Figure 2.

The image of φ1(z).

The image of f1(z).

Example 6.

If we take α=3/2, a=3/2, b=1, and c=1, then we can easily check that conditions (12) and (13) are satisfied. Furthermore, (39)Reφ2z≤0(z∈D),where (40)φ2z=D(z)Bz¯=381-z¯z2+4z-5(see Figure 3). Hence, (41)f2(z)=23F32,1,1;z=23-1+1-z/21-z3/2belongs to Co(3/2), as shown in Figure 4.

The image of φ2(z).

The image of f2(z).

3. The Pre-Schwarzian Norm Estimate for Functions in <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M208"><mml:mtext>Co</mml:mtext><mml:mo mathvariant="bold">(</mml:mo><mml:mi>α</mml:mi><mml:mo mathvariant="bold">,</mml:mo><mml:mi>A</mml:mi><mml:mo mathvariant="bold">,</mml:mo><mml:mi>B</mml:mi><mml:mo mathvariant="bold">)</mml:mo></mml:math></inline-formula>

Now, for f∈Co(α,A,B), we find the exact set of variabilities for the functional (1-|z|2)Tf(z), which gives both sharp upper and lower bounds for the pre-Schwarzian norm Tf.

Theorem 7.

Let α∈(1,2] and -1≤B<A≤1 be fixed and let f∈Co(α,A,B). Then, the set of variabilities of the functional (1-|z|2)Tf(z) is the closed disk with center (42)α+1-B2B-Aα-1z¯+α+11-z¯1-zand radius(43)12A-Bα-1.The points on the boundary of this disk are attained if and only if f is one of the functions gθ, where (44)gθ(z)=∫0z1+Beiθt(B-A)(α-1)/(2B)1-t-(1+α)dt.

Proof.

We use the characterization (4) for functions in Co(α,A,B) and the representation (45)Pf(z)=1+Azw(z)1+Bzw(z),where w:D→D¯ is a unimodular bounded analytic function. It follows that(46)Tfz=2(α+1)+(B-A)(α-1)w(z)A+Bα+3B-Azwz+A+Bα+3B-Azwz·21-z1+Bzwz-1.By a routine computation, one recognizes that (47)1-z2Tf(z)-α+1-B2B-Aα-1z¯-(α+1)1-z¯1-z=12B-Aα-1Bz¯+w(z)1+Bzw(z).Hence, the condition |w(z)|≤1 is equivalent to(48)1-z2Tfz-α+1-B2B-Aα-1z¯-α+11-z¯1-z≤12A-Bα-1.This proves the first part of the assertion in the theorem. The second part follows from the fact that |w(z)|=1 if and only if w(z)≡eiθ, θ∈[0,2π], and that the solution of the differential equation (4) in this case is given by f(z)=gθ(z). The relation between boundary points of the above circle and the extremal function becomes clear from the identity (49)1-z2Tgθ(z)-α+1-B2B-Aα-1z¯-α+11-z¯1-z=12B-Aα-1Bz¯+eiθ1+Beiθz.This completes the proof of the theorem.

Remark 8.

If we put A=1 and B=-1 in Theorem 7, then we can obtain the result in Bhowmik et al. ([2], Theorem 2.5).

From the inequality (48), we can have the following corollary.

Corollary 9.

Let f∈Co(α,A,B), α∈[1,2] and -1≤B<A≤1. Then,(50)-121-BA-Bα-1+2α+2≤Tf≤121+BA-Bα-1+2α+2.The equality holds in lower estimate for the function g0 and in upper estimate for the function gπ which are described in Theorem 7.

As a consequence of Theorem 7, we can obtain a distortion theorem for the functions in Co(α,A,B).

Theorem 10.

Let α∈(1,2] and -1≤B<A≤1. Then, for each f∈Co(α,A,B), one has(51)1-r(1/4)(1-B)(A-B)(α-1)1+r(1/4)(1+B)(A-B)(α-1)+(α+1)≤f′z≤1+r(1/4)(1-B)(A-B)(α-1)1-r(1/4)(1+B)(A-B)(α-1)+(α+1)with |z|=r<1. For each z∈D, z≠0, equality occurs if and only if f=gθ, where θ∈[0,2π).

Remark 11.

If we put A=1 and B=-1 in Theorem 10, then we can obtain the distortion theorem for the functions in Co(α) which is a result from Bhowmik et al. ([2], Theorem 2.8).

Finally, we present a characterization for functions in the class Co(α,A,B), in view of the Hadamard product.

Theorem 12.

Let 1<α≤2 and -1≤B<A≤1. Then, f∈Co(α,A,B) if and only if(52)1zfz∗-A+B+αA-αBzz21-z3-1+-4x+A+3B-αA+αBz2·1-z3-1+fz∗2α+2x+A-3B-αA-αBzz21-z3-1+2-2αx+-A-B+αA+αBz2·1-z3-1≠0for all |z|<1 and for all x with |x|=1. Equivalently, this holds if and only if (53)∑n=0∞Enzn≠0z∈D,x=1,where (54)E0=-12A-B-αA+αB,En=x-12A+B-αA+αBa12+B-xn+1(n+1)an+1+αx-12-A+B+αA+αBaaa12--x+Bnnan(n∈N)with (55)f(z)=∑n=1∞anzn(a1=1).

Remark 13.

If we put A=1 and B=-1 in Theorem 12, then we can obtain the convolution result for the functions in Co(α) which is a result from Bhowmik et al. ([2], Theorem 3.1).

Conflict of Interests

The authors declare that they have no competing interests.

AvkhadievF. G.WirthsK.-J.Concave schlicht functions with bounded opening angle at infinityBhowmikB.PonnusamyS.WirthsK.-J.Characterization and the pre-Schwarzian norm estimate for concave univalent functionsAvkhadievF. G.PommerenkeC.WirthsK.-J.Sharp inequalities for the coefficients of concave schlicht functionsAvkhadievF. G.WirthsK.-J.Convex holes produce lower bounds for coefficientsBhowmikB.PonnusamyS.WirthsK.-J.On the Fekete–Szegö problem for concave univalent functionsCruzL.PommerenkeCh.On concave univalent functionsWirthsK.-J.Julia's lemma and concave schlicht functionsCarlsonB. C.ShafferD. B.Starlike and prestarlike hypergeometric functionsMerkesE. P.ScottW. T.Starlike hypergeometric functionsPonnusamyS.Hypergeometric transforms of functions with derivative in a half planePonnusamyS.Starlikeness properties for convolutions involving hypergeometric seriesSilvermanH.Starlike and convexity properties for hypergeometric functionsSwaminathanA.Hypergeometric functions in the parabolic domainKimY. C.PonnusamyS.Sufficiency for Gaussian hypergeometric functions to be uniformly convexPonnusamyS.Close-to-convexity properties of Gaussian hypergeometric functionsPonnusamyS.VuorinenM.Univalence and convexity properties for Gaussian hypergeometric functionsRuscheweyhS.SinghV.On the order of starlikeness of hypergeometric functionsMillerS. S.MocanuP. T.