The class of functions convex in the negative direction of the imaginary axis of order (α,β)

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THE CLASS OF FUNCTIONS CONVEX IN THE NEGATIVE

DIRECTION OF THE IMAGINARY AXIS

OF ORDER

(α, β)

ADAM LECKO

Received 22 July 2001

The aim of this paper is to present an analytic characterization of the class of functions convex in the negative direction of the imaginary axis of order(α, β). The method of the proof is based on Julia’s lemma.

2000 Mathematics Subject Classiﬁcation: 30C45.

1. Introduction. In this paper, we are interested in the subclasses of functions con-vex in the negative (positive) direction of the imaginary axis of order(α, β)denoted byᏯᐂ−α,β(Ꮿᐂ+α,β).

In [1,2], the authors defined and studied the classIαof functions calledangularly accessible in the direction of the imaginary axis. Applying the method based on the Carathéodory kernel theorem, they showed an analytic characterization of functions inIα. The same classIαwith applying the Schwarz-Christoffel formulas and a method of approximation by polygons was defined and examined again in [8], where the author used the nameparallel accessible domains (functions) of orderα.

The aim of this paper is to introduce and analytically characterize functions in the classᏯᐂ−

α,β(Ꮿᐂ+α,β). In the case whenα=β=1, the results reduce to the classᏯᐂ−

(Ꮿᐂ+)of functions convex in the negative (positive) direction of the imaginary axis. These classes were distinguished as the subclasses of the class of functions convex in the direction of the imaginary axis in [6]. In [4,5], the author examined the classL0 of functions calledconvex in the direction of the negative real half-axis. To be precise, an analytic and univalent functionfin the unit diskDbelongs toL0if and only if for everyw∈f (D)the half-line{w+t:t∈[0,+∞)}is contained inf (D). Applying the Carathéodory kernel theorem the author proved, in a quite simple way, an analytic characterization of the classL0. Sinceif ∈Ꮿᐂ+_and₋_if_∈_Ꮿᐂ− _if_f_∈_L_{0, the same}

was done for the classesᏯᐂ+_and_Ꮿᐂ−_{. In [}₉_{], a new proof of analytic formulas for the}

classesᏯᐂ−(Ꮿᐂ+)based on Julia’s lemma were found. The same idea is used in this paper. At the end we notice that the classesᏯᐂ−α,αandᏯᐂ+α,αare proper subclasses ofIα.

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For eachk >0, let

Ok=

z∈D:|1−z| 2

1−|z|2< k

(2.1)

denote the disk inDcalled theoricycle. The oricycleOkis a disk inDwhose boundary circle∂Okis tangent toTatz=1.

In the proof of the main theorem, we apply the Julia lemma (see [7]; see also [3, page 56]) recalled below.

Lemma_2.1_{(Julia [}₇_]). _Let_ω_{be an analytic function in}D_with|_ω(z)|_<₁_for_z∈D_.

Assume that there exists a sequence(zn)of points inDsuch that

lim

n→∞zn=1, nlim→∞ω

zn

=1, (2.2)

lim n→∞

1−ωzn 1−zn =

λ <∞. (2.3)

Then

1−ω(z)2

1−ω(z)2≤λ

|1−z|2

1−|z|2, z∈D, (2.4)

and hence, for everyk >0,

ωOk

⊂Oλk. (2.5)

Remark_2.2. _Since

1−ωz

1−|z| ≥

1−ω0

1+ω0, z∈D, (2.6)

for every functionωanalytic inDwith|ω(z)|<1 forz∈D, the constantλdeﬁned in (2.2) is positive (see [3, page 43]).

3. Convexity in the negative direction of the imaginary axis of order(α, β). We start with notation. Forw∈Candθ∈[0,2π ), let

l[w, θ]=w+teiθ_:_t_∈_[₀_,_+∞₎_. _(3.1)

ForA, B⊂Candw∈C, let

A±B= {u±v∈C:u∈A∧v∈B}, A+w=A+{w}. (3.2)

For ﬁxedα, β∈[0,1], let

A(α, β)=

z∈C:−(1−α)π

2 ≤argz≤(1−β)

π

2 ∪{0}. (3.3)

Clearly,A(0,0)= {z∈C: Rez≥0}andA(1,1)=l[0,0]=[0,∞). Notice thatA(α, β), whenα≠1 orβ≠1, is a closed convex sector with the half-linesl[0, (1−β)π /2]and

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Definition3.1. Fixα, β∈[0,1]. A domainΩ⊂C,Ω≠ C, is calledconvex in the negative direction of the imaginary axis of order (α, β)if and only ifw+iA(α, β)is contained inΩfor everyw∈Ω. The set of all such domains will be denoted byᐆ−

α,β.

Definition_3.2. _LetᏯᐂ−_α,β_{denote the class of all functions}_f∈᏿_{such that}_{f (}D₎∈

ᐆ−

α,β. Functions inᏯᐂ−α,βwill be calledconvex in the negative direction of the imaginary axis of order(α, β).

The classᐆ−1,1denoted for short byᐆ−and the corresponding classᏯᐂ−1,1denoted byᏯᐂ−_{contain domains and functions called}_{convex in the negative direction of the} imaginary axis, respectively.

Lemma_3.3. _If₀≤_α₁≤_α₂≤₁_,₀≤_β₁≤_β₂≤₁_{, and}Ω∈ᐆ−_α

1,β1, thenΩ∈ᐆ

−

α2,β2.

Sinceᐆ−

α,β⊂ᐆ−for allα, β∈[0,1], every domain inᐆ−α,βis simply connected. It is obvious that, for everyf ∈Ꮿᐂ−

α,β, there are some points onTwhich “corre-spond” to inﬁnity lying on the boundary off (D). In what follows, we will use a kind of the boundary normalization for everyf∈Ꮿᐂ−

α,βby saying thatz=1 corresponds to∞ ∈∂f (D). Since, in general, we cannot extendf onT, in order to be precise, we will apply the notion of prime ends to formulate this normalization. Below we con-struct a prime endp_∞(Ω)for everyΩ∈ᐆ−

α,βand next using the prime end theorem we associatez=1 withp∞(Ω).

Since for eachα, β∈[0,1],Ꮿᐂ−α,β⊂Ꮿᐂ−, we can construct for every domainΩin ᐆ−

α,β, a prime endp∞(Ω)in this way like in [9].

Construction of a prime end for the domain convex in the negative

direction of the imaginary axis. Whenα=β=1 the detailed construction was

presented in [9]. The same construction is valid forα≠1 or β≠1 since Ꮿᐂ−α,β⊂ Ꮿᐂ− _{for all} _{α, β}_∈_[₀_,₁_]_{. But in what follows we need some notations used in the}

construction, so we recall it again. LetΩ∈ᐆ−

α,β.

(1) Assume ﬁrst thatΩis neither a vertical strip nor a half-plane with the boundary straight line parallel to the imaginary axis. Then there existsw0∈∂Ωsuch that(w0+

iA(α, β))\{w0}lies inΩ. Hence(l[w0, π /2]\{w0})⊂Ω. For eacht∈(0,∞), we denote

C(t)= {w∈C:|w−w0| =t}. It is clear thatΩ∩C(t)≠∅for everyt∈(0,∞). By [10, Proposition 2.13, page 28], for each t∈(0,∞)there are countably many crosscuts

Ck(t)⊂C(t),k∈N, ofΩeach of which is an arc of the circleC(t). ByΩ0(t)⊂Ωwe denote the component ofΩ\C(t)containing the half-linel[w0+it, π /2]\ {w0+it} and byQ(t)∈k∈NCk(t) we denote the crosscut containing the point w0+it. So

Q(t)⊂∂Ω0(t). Let now(tn)be a strictly increasing sequence of points in(0,∞)such that limn→∞tn= ∞and let(Q(tn))be the corresponding sequence of crosscuts ofΩ. It is easy to observe that

(i) Q(tn)∩Q(tn+1)= ∅for everyn∈N; (ii) Ω0(tn+1)⊂Ω0(tn)for everyn∈N;

(iii) diam#Q(tn)→0 asn→ ∞, where diam#Bmeans the spherical diameter of the setB⊂C.

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The equivalence class of the null chain(Cn)deﬁnes the prime end denoted byp∞(Ω).

We can also show that inﬁnity is a unique principal point of the prime endp∞(Ω).

(2) (a) LetΩbe a vertical strip of widthd >0. Clearly, this is possible only whenα=

β=1. Letw0∈∂Ω. For eacht∈(d,∞), setC(t)= {w∈C:|w−w0| =t}. It is clear that Ω∩C(t)≠∅for everyt∈(d,∞). Observe thatΩ(t)is a sum of two disjoint circular arcs, denoted by Q+(t) and Q−(t). Let Q+(t) be the circular arc which lies above

Q−(t). Precisely,Q+(t)cuts the boundary straight lines ofΩat two points:w1(t)and

w2(t), and together with half-linesl[w1(t), π /2]andl[w2(t), π /2]is a boundary of a domain denoted byΩ+_(t)_{. Moreover,}_Ω+_(t)_⊂_Ω_and_Ω+_(t)_∩_Int_C(t)_{= ∅}_.

Let now(tn)be a strictly increasing sequence of points in(d,∞)such that limn→∞tn = ∞, and let(Q+(tn))be the corresponding sequence of crosscuts ofΩ. It is easy to observe that the conditions (i)–(iii) listed in part (1) are fulﬁlled. Therefore (Cn+)=

(Q+(tn))forms a null chain ofΩ. The null chain(Cn+)is independent of the choice of the sequence(tn).

The equivalence class of the null chain (Cn+) deﬁnes the prime end denoted by

p_∞+(Ω). We can also say that inﬁnity is a unique principal point of the prime endp_∞+(Ω). In a similar way the sequence(Q−(tn))is a null chain which represents the second prime endp_∞−(Ω), diﬀerent fromp_∞+(Ω).

For the next considerations, the prime endp+_∞(Ω)will be denoted byp_∞(Ω). (b) Let nowΩbe a half-plane with the boundary straight line parallel to the imaginary axis. Letw0∈∂Ω, and for each t∈(0,∞), letC(t)= {w∈C:|w−w0| =t}. It is clear thatQ(t)=Ω∩C(t) is a halfcircle for every t >0. Repeating considerations similar to those above we see that the sequence(Cn)=(Q(tn)), for an arbitrary strictly increasing sequence(tn)of points in(0,∞)such that limn→∞tn= ∞, forms a null chain ofΩwhich represents a prime end denoted byp_∞(Ω).

In this way, we construct for everyΩ∈ᐆ−

α,β, in a unique way, a prime endp∞(Ω). We can also show that inﬁnity is a unique principal point of the prime endp∞(Ω).

Therefore, the following proposition follows.

Proposition3.4. For everyΩ∈ᐆ−_α,β,α, β∈[0,1], the prime endp_∞(Ω)is of the

first or of the second kind.

Let f ∈Ꮿᐂ−α,β and Ω=f (D). By the prime end theorem there exists a bijective mapping ˆf of the unit circleTonto the set of all prime ends ofΩ(see [10, page 30]). Hence there is a uniqueζ_∞∈Tsuch that p_∞(Ω)=_{f (ζ}ˆ _∞₎_{. We can also show that} inﬁnity is a unique principal point of the prime endp∞(Ω).

If nowf∈Ꮿᐂ−

α,β, then we can writep∞(Ω)=f (ζˆ ∞)for uniqueζ∞∈T.

4. An analytic characterization of the class of function convex in the negative direction of the imaginary axis of order(α, β). In the proof of the main theorem, which analytically characterizes the classᏯᐂ−α,β, we need the following lemma which was proved in [9] in an easy way.

Lemma_4.1. _{Every sequence}_(a_n₎_{of positive numbers with}

lim n→∞

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has a convergent subsequence(ank)and

0≤lim

k→∞ank=a≤1. (4.2)

Now we prove the theorem which says that every functionf∈Ꮿᐂ−

α,β, withp∞(f (D))

=_{f (}ˆ₁₎_{, preserves convexity in the negative direction of the imaginary axis of order}

(α, β)on every oricycleOk.

Theorem4.2. Letα, β∈[0,1]andf∈᏿. Thenf∈Ꮿᐂ−_α,βandp_∞(f (D))=f (ˆ1),

if and only iff (Ok)∈ᐆ−α,βfor everyk >0.

Proof. (1) Assume that f ∈Ꮿᐂ−_α,β and ζ_∞= 1 corresponds to the prime end

p_∞(f (D)).For eachu∈A(α, β), let

ωu(z)=f−1

f (z)+iu, z∈D. (4.3)

Sincef (D)∈ᐆ−

α,β,f (z)+iu∈f (D)for everyu∈A(α, β)andz∈D. Hence, from the univalence off, it follows thatωuis well deﬁned.

Fixu∈A(α, β)and letΩ∈ᐆ−α,β.

We select two points:w0∈∂Ωandw1∈Ω, in the following way. IfΩis not a vertical strip or a half-plane with the boundary straight line parallel to the imaginary axis, then there existsw0∈∂Ωsuch that(w0+iA(α, β))\{w0}lies inΩ. Since(w0+iu)∈(w0+

iA(α, β)), the half-linelstarting fromw0and going throughulies inw0+iA(α, β). Consequently,(l\{w0})⊂Ω. Fixw1∈l\{w0}.

In the case whenΩis a vertical strip or a half-plane with the boundary straight line parallel to the imaginary axis, letw1∈Ωbe arbitrary andw0∈∂Ωbe such that Imw1=Imw0.

Assume now that, forΩ=f (D), the pointsw0andw1are chosen as above. Consider the sequence (wn)=(w1+i(n−1)u)of points inl\ {w0}and the corresponding sequence(zn)=(f−1(wn))of points inD.

With a notation as in the construction of a prime, end let C(tn) = {w ∈ C: |w−w0| = |wn−w0|}, wheretn= |wn−w0|, and letQ(tn)⊂C(tn), forn∈N, denote the crosscut off (D)containingwn. From the method of choosingw0andw1, we see that the conditions (1)–(3) are satisﬁed and(Q(tn))is a null chain representing the prime endp_∞(f (D)). By the prime end theorem(f−1_(Q(t

n)))is a null-chain inDthat separates the origin fromζ∞=1 for largen. Sincezn=f−1(wn)∈f−1(Q(tn))and diamf−1_(Q(t

n))→0 forn→ ∞, we conclude that limn→∞zn=1. Observe that

ωu

zn

=f−1wn+iu

=zn+1. (4.4)

Let now

an=

1−ωu

zn 1−zn

, n∈N. (4.5)

Hence

an=

1−ωu

zn 1−zn =

1−zn+1 1−zn

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for alln∈N. Consequently,

lim n→∞

a1a2···an=lim n→∞

1−z2 1−z1

1−z3 1−z2···

1−zn 1−zn−1

1−zn+1 1−zn

=lim n→∞

1−zn+1 1−z1 =

0.

(4.7)

ByLemma 4.1, there exists a convergent subsequence(an_k)of the sequence(an)such that

0≤lim

k→∞ank=λ(u)≤1. (4.8)

Hence we conclude that, for eachu∈A(α, β), there exists a convergent subsequence

(zn_k)of the sequence(zn)such that

lim k→∞

1−ωu

znk 1−zn_k =

λ(u)≤1. (4.9)

In view ofRemark 2.2,λ(u) >0 for everyu∈A(α, β). By this way,ωu satisﬁes the assumptions of the Julia lemma withλ(u)∈(0,1]. Hence

ωuOk⊂Oλ(u)k⊂Ok (4.10)

for everyu∈A(α, β)andk >0. This yieldsf−1_{(f (}_O

k)+iu)⊂Ok, sof (Ok)+iu⊂

f (Ok)for everyu∈A(α, β). Thereforef (Ok)∈ᐆ−α,βfor everyk >0.

(2) Now assume thatf (Ok)∈ᐆ−α,βfor everyk >0. Since∞ ∈∂f (Ok)for everyk >0 and

f (D)= k>0

fOk

, (4.11)

∞ ∈∂f (D)andf (D)∈ᐆ−α,β. Observe also that there exists a prime endp∞(f (D)) which corresponds to some pointζ_∞∈T. We need to show thatζ_∞=1.

To this end, letk >0 be ﬁxed and suppose thatζ∞≠1.

Let(Q(tn))be an arbitrary sequence of crosscuts off (D)which represents the prime endp_∞(f (D))corresponding in a unique way to a pointζ_∞∈T, that is,(Q(tn)) is a null-chain off (D). By the prime end theorem(f−1_(Q(t

n)))is a null-chain that separates inDthe origin andζ∞for largen. Sinceζ∞≠1 and diamf−1(Q(tn))→0 forn→ ∞we see that

f−1_Q_t

n∩Ok= ∅, (4.12)

for largen.

On the other hand,f (Ok)∈ᐆ−α,β, which implies thatQ(tn)∩f (Ok)≠∅for large

n∈N.This contradicts (4.12) and shows thatζ∞=1 andp∞(f (D))=f (ˆ1). The proof

of the theorem is ﬁnished.

Using Theorem 4.2we ﬁnd an analytic characterization of functions in the class Ꮿᐂ−

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Theorem4.3. Letα, β∈[0,1]. Iff∈Ꮿᐂ−_α,βandp_∞(f (D))=f (ˆ1), then

−βπ

2 ≤arg

−i(1−z)2_f_(z)_≤_απ

2, z∈D. (4.13)

Proof. _LetΩ=_{f (}D₎_{. The case}_α=_β=_{1 is well known and can be found in [}₅_,₆_,₉_].

Assume thatα≠1 orβ≠1. This means thatA(α, β)is a closed convex sector which does not reduce to the half-linel[0,0]. Now we prove that (4.13) is true for all points onγk=∂Ok\{1}for everyk >0. We use the following parametrization ofγk:

γk:z=z(θ)=

1+keiθ

1+k , θ∈(0,2π ). (4.14)

LetΓk=f (γk), sinceγkis positively oriented, so isΓk. For eachz∈γkwe denote by

τ(z)the tangent vector toΓkatw=f (z), that is,

τ(z)=z(θ)fz(θ), (4.15)

wherez=z(θ)is given by (4.14). Since

1−z(θ)2= k2

(1+k)2

1−eiθ2

=4ksin2(θ/2)

k+1 z

_(θ)i

=2 Re1−z(θ)z(θ)i, θ∈(0,2π ),

τ(z)= −i(_{2 Re}1−z)_{₁2₋f_z(z)_} , z∈γk.

(4.16)

Let V denote the closed convex sector with vertex at w and with the half-lines

l[w, απ /2]andl[w,2π−βπ /2]as its arms. This means that(w+iA(α, β))∪Vforms a closed half-plane containing the half-linel[w, απ /2].

Fix k >0. By Theorem 4.2, f (Ok)∈ᐆ−α,β for every k > 0. Therefore by an easy observation we see thatw+iA(α, β)⊂f (Ok). Hence it follows that the tangent line toΓkatwcannot intersect the interior of the sectorw+iA(α, β). This implies that

τ(z)lies inV. Consequently, in view of (4.16), we have

−βπ

2 ≤arg

τ(z)=arg−i(1−z)2_f_(z)_≤_απ

2 (4.17)

forz∈γk. Askwas arbitrary, this is true inD.

Now we prove the converse theorem.

Theorem 4.4. Let α, β∈[0,1]. If f ∈Ꮽand (4.13) is true, then f ∈Ꮿᐂ−_α,β and

p∞(f (D))=f (ˆ1).

Proof. (1) Suppose that there existsz₀∈Dsuch that the equality in the left-hand

side of (4.13) holds. Then it holds in the whole diskD, that is, there exists a positive real numberasuch that

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This is satisﬁed only for

f (z)=b+aie₁−₋iβπ /2_z , z∈D, (4.19)

whereb∈C.

In a similar way, if the equality in the right-hand side of (4.13) holds for somez0∈D, then it holds only for

f (z)=b+aieiαπ /2

1−z , z∈D, (4.20)

whereb∈C.

Particularly, ifα=β=0, then (4.13) is true only for

f (z)=b+ ai

1−z, z∈D, (4.21)

whereb∈Canda∈R\{0}.

Functions (4.19) and (4.20) mapDonto half-planes and a simple geometric viewing shows that they are elements ofᏯᐂ−α,βwithp∞(f (D))=f (ˆ1).

(2) Suppose that in (4.13) strong inequalities holds. Sincef∈Ꮿᐂ−_,_f _{is univalent}

inD(see [5,4,6]). We show thatf (D)∈ᐆ−

α,β. Suppose, on the contrary, thatf (D)∈ᐆ−

α,β. ByTheorem 4.2, there existsk >0 such thatf (Ok)∈ᐆ−α,β. This means that(w0+iA(α, β))\ {w0}is not contained inf (Ok) for somew0∈f (Ok).

Suppose that

Γk∩l

w0, π−β

π

2

≠∅. (4.22)

Thus there existsw1∈Γk∩l[w0, π−βπ /2],w1≠w0, such that the segment[w0, w1) lies inf (Ok). Letτ(z1)be the tangent vector toΓkatw1=f (z1), wherez1∈γk. LetV denote the closed convex sector with vertex atw1and with the half-linesl[w1, απ /2] and l[w1,2π−βπ /2]as its arms. This means that (w1+iA(α, β))∪V is a closed half-plane containing the half-linel[w1, απ /2]. LetH be the complementary closed half-plane. Henceτ(z1)lies inHwhich means that

argτz1

∈

−π ,−βπ

2

∪

π−βπ

2, π

, (4.23)

contrary to (4.13).

In a similar way we obtain a contradiction assuming that

Γk∩l

w0, α

π

2

≠∅. (4.24)

This ends the proof.

Remark4.5. Forα=β=1, Theorems4.3and4.4show the well-known analytic

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The following theorems are immediate consequences of Theorems 4.3 and 4.4 by applying them to the function f (z)=g(e−iµ_z)_, _z_∈ _D_{, where} _g_∈ _Ꮿᐂ−

α,β and

p_∞(g(D))=g(ˆ 1).

Theorem_4.6. _Let_{α, β}∈_[₀_,₁_]_{. If}_f∈Ꮿᐂ_α,β− _and_p_∞_{(f (}D₎₎=_{f (e}ˆ iµ₎_,_µ∈R_{, then}

−βπ

2 ≤arg

−ieiµ₁₋_e−iµ_z2

f(z)≤απ

2, z∈D. (4.25)

Theorem_4.7. _Let_{α, β}∈_[₀_,₁_]_{. If}_f∈Ꮽ_{and (}_4.25_{) is true for}_µ∈R_{, then}_f∈Ꮿᐂ−_α,β

andp∞(f (D))=f (eˆ iµ).

5. Convexity in the positive direction of the imaginary axis of order(α, β). The results presented inSection 4can be applied at once to the functions calledconvex in the positive direction of the imaginary axis of order(α, β).

Definition_5.1. _Fix_{α, β}∈_[₀_,₁_]_{. A domain}Ω⊂C_, Ω≠ C_{, will be called}_convex

in the positive direction of the imaginary axis of order (α, β)if and only if the sector

w−iA(α, β)is contained inΩfor everyw∈Ω. The set of all such domains will be denoted byᐆ+α,β.

Definition5.2. LetᏯᐂ+_α,βdenote the class of all functionsf∈᏿such thatf (D)∈

ᐆ+

α,β. Functions in the classᏯᐂ+α,βwill be calledconvex in the positive direction of the imaginary axis of order (α, β).

Sincef∈Ꮿᐂ+

α,βif and only if−f∈Ꮿᐂ−α,βwe have the following theorems.

Theorem_5.3. _Let_{α, β}∈_[₀_,₁_]_{and let}_f∈᏿_{. Then}_f∈Ꮿᐂ+_α,β_and_p_∞_{(f (}D₎₎=_{f (}ˆ₁₎_,

if and only iff (Ok)∈ᐆ+α,βfor everyk >0.

Theorem_5.4. _Let_{α, β}∈_[₀_,₁_]_{. If}_f∈Ꮿᐂ_α,β+ _and_p_∞_{(f (}D₎₎=_{f (e}ˆ iµ₎_,_µ∈R_{, then}

−βπ

2 ≤arg

ieiµ₁₋_e−iµ_z2

f(z)≤απ

2, z∈D. (5.1)

Theorem5.5. Letα, β∈[0,1]. Iff∈Ꮽand (5.1) is true forµ∈R, thenf∈Ꮿᐂ+_α,β

andp∞(f (D))=f (eˆ iµ).

References

[1] C. Burniak,Zastosowanie metody homotopii, pewego kryterium jednolistno´sci i innych metod w badaniu pewnych funkcji holomorficznych w kole jednostkowym, Ph.D. thesis, Univ. Marie Curie Skłodowska, Lublin, 1982.

[2] C. Burniak and J. Godula,On functions angularly accessible in the direction of the imagi-nary axis, Ann. Univ. Mariae Curie-Skłodowska Sect. A35(1981), 43–51.

[3] C. Carathéodory,Conformal Representation, Cambridge University Press, Cambridge, 1963.

[4] K. Ciozda,O Klasie funkcji wypukłych w ujemnym kierunku osi rzeczywistej, jej pod-klasach i podstawowych własno´sciach, Ph.D. thesis, Univ. Marie Curie Skłodowska, Lublin, 1978.

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[6] W. Hengartner and G. Schober,On Schlicht mappings to domains convex in one direction, Comment. Math. Helv.45(1970), 303–314.

[7] G. Julia, Extension nouvelle d’un lemme de Schwarz, Acta Math. 42 (1920), 349–355 (French).

[8] W. Koepf,Parallel accessible domains and domains that are convex in some direction, Partial Diﬀerential Equations with Complex Analysis, Pitman Res. Notes Math. Ser., vol. 262, Longman Scientiﬁc & Technical, Harlow, 1992, pp. 93–105.

[9] A. Lecko,On the class of functions convex in the negative direction of the imaginary axis, to appear in J. Austral. Math. Soc.

[10] C. Pommerenke,Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin, 1992.

Adam Lecko: Department of Mathematics, Technical University of Rzeszów, ul. W. Pola2,35-959Rzeszów, Poland