SOME PROPERTIES OF LINEAR OPERATORS DEFINED BY
HYPERGEOMETRIC FUNCTIONS
Z. Orouji and R. Aghalary
Department of Mathematics, Faculty of Science, Urmia University, Urmia, Iran
e-mails: [email protected], [email protected], [email protected]
(Received 18 July 2017; after final revision 15 December 2017;
accepted 13 April 2018)
This paper is concerned with the connection of certain operatorsΦb,c, defined by hypergeometric functions, with theα-logarithmic Bloch spacesBlog,αof analytic functions.
Key words : Fractional derivative;α-logarithmic Bloch spaces; nontangential limits; Pochham-mer symbol; uniformly locally univalent.
1. INTRODUCTION
LetH(D)denote the class of analytic functionsf on the open unit discD={z∈C:|z|<1}. Also denote byAthe subclass ofH(D)consisting of functions normalized by
f(z) =z+
∞ X
n=2
anzn.
Forα >0, theα-logarithmic Bloch spaceBlog,αis the Banach space of those functionsf ∈ H(D) which satisfy
kf klog,α:=|f(0)|+ sup z∈D
(1− |z|2)
µ
log 2 1− |z|2
¶α
|f0(z)|<∞.
For simplicity, the spaceBlog,1will be denoted byBlog.
The logarithmic Bloch spaces were studied by many authors (see for example [3] and [4]). A typical example of an unbounded univalent function in the spaceBlogis the function
f(z) = log log 2
Let us mention also that the logarithmic Bloch spaces play a basic role studying the boundedness of Hankel and Toeplitz operators in Bergman and Bloch-type spaces (see. e.g., [1] and [12]).
Various operators of fractional calculus (that is, fractional integral and fractional derivative) have been studied in the literature (cf. [5, 9-11]). We use the following definition used recently by Srivas-tava and Owa [10]. The fractional integral of orderλis defined, for a functionf, by
Dz−λf(z) = 1 Γ(λ)
Z z
0
f(ζ)
(z−ζ)1−λ dζ, (λ >0),
wheref is an analytic function in a simply-connected region of thez-plane containing the origin, and the multiplicity of(z−ζ)λ−1is removed by requiringlog(z−ζ)to be real whenz−ζ >0.
It is easy to check that forf(z) =z+P∞n=2anzn∈ Awe have Dz−λf(z) = 1
Γ(2 +λ)z λ+1+
∞ X
n=2
Γ(n+ 1) Γ(λ+n+ 1)anz
n+λ, (z∈D). (1.1)
Also the fractional derivative of orderλis defined, for a functionf, by Dλzf(z) = 1
Γ(1−λ) d dz
Z z
0
f(ζ)
(z−ζ)λ dζ, (0≤λ <1),
wherefis a above, and the multiplicity of(z−ζ)−λis removed, as in previous definition. Also it is interesting to note that forf(z) =z+P∞n=2anzn∈ Awe have
Dzλf(z) = 1 Γ(2−λ)z
1−λ+X∞ n=2
Γ(n+ 1) Γ(n+ 1−λ)anz
n−λ, (z∈D). (1.2)
Jung et al. [5] investigated the linear operator=αβ which is defined by =αβf(z) = Γ(α+β+ 1)
Γ(α+ 1)Γ(β+ 1) α zβ
Z z
0
µ
1− t z
¶α−1
tβ−1f(t)dt, (f ∈ H(D), z∈D)
α >0;β >−1;
and they proved forf ∈ Awith Ref0(z) > 0,=αβf is bounded at least forα > 1andβ > −1. In view of ([5], Lemma 4), for functionf(z) =z+P∞n=2anzn∈ Awe have
=αβf(z) =z+Γ(α+β+ 1) Γ(β+ 1)
∞ X
n=2
Γ(β+n) Γ(α+β+n) anz
n, (α >0, β >−1). (1.3)
Definition 1.1 — Forf(z) =P∞n=0anznwithf ∈ H(D), we define Φb,cf(z) =
∞ X
n=0
(b)n (c)nanz
Hereb, c are complex numbers such that c 6= 0,−1,−2, ...,(b)0 = 1for b 6= 0, and for each
positive integern,(b)n=b(b+ 1)(b+ 2)...(b+n−1)is the Pochhammer symbol.
The above series converges absolutely for allz∈D,and hence represents an analytic function in the unit discD. It should be remarked that by specializing the parametersb, cand using (1.1), (1.2) and (1.3) one can obtain that
D−zλf(z) = zλ
Γ(1 +λ)Φ1,1+λf(z), (z∈D, λ >0), (1.4)
Dλzf(z) = z−λ
Γ(1−λ)Φ1,1−λf(z), (z∈D,0≤λ <1), (1.5) and
=αβf(z) = α+β
β Φβ,α+βf(z), (z∈D, α >0, β >0), (1.6) wheref ∈ A.
It is natural to ask that for which functionsf and parametersb, c, Φb,cf ∈ Blog,α(α > 0). The main purpose of this paper is finding conditions on the parametersb, csuch that whenf is chosen from suitable classes,Φb,cf ∈ Blog,α(α >0).
Remark 1.1 : It is easy to see that the function g(x) = (1−x)
³
log1−2x
´α
(0 < x < 1)is bounded and decreasing for0≤α <ln 2.
Proving our results we shall use the following well known result
Z 1
0
tp−1(1−t)q−1dt= Γ(p)Γ(q) Γ(p+q)
where Rep >0and Req >0.
2. MAINRESULTS
In the following Theorem we prove:
Theorem 2.1 — Letb, cbe complex numbers such that−1 <Rec <Reb. Also letf ∈ H(D) andΦb,cf ∈ Blog,αwhere0≤α <ln 2. Thenf ∈ Blog,α. Moreover
kfklog,α≤ |f(0)|+ |Γ(b)|Γ(c)|Γ(Re(b||Γ(b−−c)c))Γ(Re c|Γ(Re b+ 1)+ 1)kΦb,cfklog,α.
forz∈Dwe have
Z 1
0
(1−t)b−c−1tc−1Φb,cf(tz)dt= Γ(c) Γ(b)
Z 1
0
(1−t)b−c−1tc−1
∞ X
n=0
Γ(b+n) Γ(c+n)ant
nzndt
= Γ(c) Γ(b)
∞ X
n=0
Γ(b+n) Γ(c+n)anz
n
Z 1
0
(1−t)b−c−1tn+c−1dt
= Γ(c)Γ(b−c) Γ(b)
∞ X
n=0
anzn
= Γ(c)Γ(b−c)
Γ(b) f(z). (2.1)
By Remark 1.1 and relation (2.1), we obtain (1− |z|2)
µ
log 2 1− |z|2
¶α
|f0(z)|
≤ |Γ(b)| |Γ(c)||Γ(b−c)|
Z 1
0
(1−t)Re(b−c)−1tRe c(1− |z|2)
µ
log 2 1− |z|2
¶α
|(Φb,cf)0(tz)|dt
≤ |Γ(b)| |Γ(c)||Γ(b−c)|
Z 1
0
(1−t)Re(b−c)−1tRe c(1− |tz|2)
µ
log 2 1− |tz|2
¶α
|(Φb,cf)0(tz)|dt
≤ |Γ(b)|kΦb,cfklog,α |Γ(c)||Γ(b−c)|
Z 1
0
(1−t)Re(b−c)−1tRe cdt
<∞, (z∈D),
since−1<Rec <Reb. Thereforef ∈ Blog,αand
kfklog,α≤ |f(0)|+|Γ(b)|Γ(c)|Γ(Re(b||Γ(b−−c)c))Γ(Re c|Γ(Re b+ 1)+ 1)kΦb,cfklog,α.2
By puttingb = 1andc = 1−λin the Theorem 2.1 and using (1.5) we obtain the following corollary:
Corollary 2.1 — Letf ∈ AandzλDλzf ∈ Blog,αwhere0 ≤ λ < 1 and0 ≤ α <ln 2. Then f ∈ Blog,αandkfklog,α= (1−λ)kzλDzλfklog,α.
A functionf ∈ H(D)is called uniformly locally univalent if there exists a constantρ >0such thatf is univalent on the hyperbolic disc|(z−a)/(1−az)¯ |<tanhρof radiusρfor everya∈D. It is known that a non-constant analytic functionf is uniformly locally univalent if and only if the norm
kf00/f0 k= sup z∈D
(1− |z|2)
¯ ¯ ¯ ¯f
00(z)
of the pre-Schwarzian derivativef00/f0offis finite. Let
B(λ) ={f ∈ H(D);kf00/f0k≤2λ}.
Kim and Sugawa [6, 7] investigated various properties of the functions which belong to the class B(λ). In the next theorem we prove:
Theorem 2.2 — Letλ > 0and letb, cbe complex numbers such that−1 <Reb <Rec− λ2. Also suppose that f(z) = P∞n=0anzn is uniformly locally univalent withk f00/f0 k≤ λ. Then Φb,cf ∈ Blog,αfor0≤α <ln 2.
PROOF: Let|z|=r (0< r <1)andkf00/f0 k≤λ. Then we have log
¯ ¯ ¯ ¯f
0(z)
f0(0) ¯ ¯ ¯ ¯≤
¯ ¯ ¯ ¯logf
0(z)
f0(0) ¯ ¯ ¯ ¯
=
¯ ¯ ¯ ¯
Z z
0
f00(w)
f0(w)dw ¯ ¯ ¯ ¯
≤r
Z 1
0
¯ ¯ ¯ ¯f
00(tz)
f0(tz) ¯ ¯ ¯ ¯dt
≤r
Z 1
0
λ 1−r2t2dt
=λlog
r
1 +r 1−r.
This implies
|f0(z)| ≤ |f0(0)|
µ
1 +r 1−r
¶λ
2
, (|z|=r <1). (2.2) On the other hand, we have
Φb,cf(z) = Γ(c)Γ(b)
∞ X
n=0
Γ(b+n) Γ(c+n).
c+n c+nanz
n
= Γ(c) Γ(b)Γ(c−b+ 1)
∞ X
n=0
Γ(b+n)Γ(c−b+ 1)
Γ(c+n+ 1) (c+n)anz n
= Γ(c) Γ(b)Γ(c−b+ 1)
Z 1
0
tb−1(1−t)c−b
∞ X
n=0
(c+n)anzntndt
= Γ(c) Γ(b)Γ(c−b+ 1)
Z 1
0
tb−1(1−t)c−b(cf(tz) +tzf0(tz))dt. (2.3)
Seth(z) =cf(z) +zf0(z). Sincekf00/f0 k≤λ, we obtain|f00(z)| ≤ λ1|−|f0(zz|)2|(z∈D),and so |h0(z)| ≤ |f0(z)|
µ
λ|z|
1− |z|2 +|c|+ 1
¶
Then in view of (2.3) and (2.4) we have |(Φb,cf)0(z)| ≤ |Γ(c)|
|Γ(b)||Γ(c−b+ 1)|
Z 1
0
tRe b(1−t)Re(c−b)|f0(tz)|
µ
λt|z|
1−t2|z|2 +|c|+ 1
¶
dt
and by relation (2.2), for|z|=r(0< r <1),we obtain (1− |z|2)
µ
log 2 1− |z|2
¶α
|(Φb,cf)0(z)|
≤ |Γ(c)|
|Γ(b)||Γ(c−b+ 1)|
Z 1
0
tRe b(1−t)Re(c−b)(1− |z|2)
µ
log 2 1− |z|2
¶α
|f0(0)|
×
µ
1 +tr 1−tr
¶λ
2 µ λt|z|
1−t2|z|2 +|c|+ 1
¶
dt
≤ |Γ(c)||f0(0)|
|Γ(b)||Γ(c−b+ 1)|(log 2) α
Z 1
0
tRe b(1−t)Re(c−b)
µ
2 1−t
¶λ
2 µ λt
1−t+|c|+ 1
¶
dt
≤ |Γ(c)||f
0(0)|2λ2
|Γ(b)||Γ(c−b+ 1)|λ(log 2) α
Z 1
0
tRe b+1(1−t)Re(c−b)−λ2−1
+ |Γ(c)||f0(0)|2
λ
2
|Γ(b)||Γ(c−b+ 1)|(|c|+ 1)(log 2) α
Z 1
0
tRe b(1−t)Re(c−b)−λ2
<∞
since−1<Reb <Rec−λ2 and soΦb,cf ∈ Blog,α.
By puttingb = 1andc = 1 +λin the Theorem 2.2 and using (1.4) we obtain the following corollary:
Corollary 2.2 — Letf(z) =z+P∞n=2anzn ∈B(p)and0< p < λ. Thenz−λDz−λf ∈ Blog,α for0≤α <ln 2.
Also by puttingb=βandc=γ+βin the Theorem 2.2 and using (1.6) we obtain:
Corollary 2.3 — Letf(z) =z+P∞n=2anzn ∈ B(p)and0 < p < γ. Then=γβf ∈ Blog,α for 0≤α <ln 2andβ >0.
Theorem 2.3 — Let0≤α <ln 2and let0< b < c−1. Also supposef(z) =P∞n=1anznand g(z) =P∞n=1 an
nc−bznbe analytic inDwhere{an}is a bounded sequence inC. Theng ∈ Blog,α if and only if Φb,cf ∈ Blog,α.
PROOF : From Stirlings formula, we have the following asymptotic expansion for the gamma function(|argz| ≤π−², ² >0)([8], p. 88):
Γ(z)≈e−zzz
r
2π z
µ
1 + 1 12z +
1
288z2 +...
¶
Hence we obtain (asn→ ∞) Γ(b+n)
Γ(c+n) ≈e c−b
µ
b+n c+n
¶b+n−1 2
(c+n)b−c
µ
1 + c−b
12(b+n)(c+n) +...
¶
. (2.5)
Note that (asn→ ∞)
µ
b+n c+n
¶b+n−1 2
≈e−(c−b)
µ
1 + c−b 2(b+n) +...
¶
(2.6) and
(c+n)b−c=nb−c
µ
1−−c n
¶−(c−b)
=nb−c
µ
1 +−c(c−b) n +...
¶
. (2.7)
Therefore by relations (2.5) to (2.7), we have Γ(b+n)
Γ(c+n) ≈n b−cX∞
i=0
Ain−i (n→ ∞) (2.8)
where theAiare constants depending onb, cwithA0 = 1. From (2.8) we obtain
Φb,cf(z) = Γ(c) Γ(b)
∞ X
n=1
Γ(b+n) Γ(c+n)anz
n ≈ Γ(c) Γ(b) ∞ X n=1
nb−c
(∞
X
i=0
Ain−i
)
anzn,
and so
Γ(b)
Γ(c)Φb,cf(z) =g(z) +
∞ X
n=1
nb−cO
µ
1 n
¶
anzn (2.9)
whereg(z) =P∞n=1 an
nc−bzn. Sincean=O(1), we have ¯ ¯ ¯ ¯ ¯ ∞ X n=1
nb−c+1O
µ
1 n
¶
anzn−1
¯ ¯ ¯ ¯ ¯≤ µ max n∈N|an|
¶X∞
n=1 ¯ ¯ ¯ ¯O µ 1 nc−b
¶¯¯ ¯ ¯
<∞
since0< b < c−1and so there existsN >0such that
¯ ¯ ¯ ¯ ¯ ∞ X n=1
nb−c+1O
µ
1 n
¶
anzn−1
¯ ¯ ¯ ¯
¯< N, (z∈D). (2.10)
Now, supposeg∈ Blog,αwhere0≤α <ln 2. From (2.9) we have (Φb,cf)0(z) = Γ(c)Γ(b)
Ã
g0(z) +
∞ X
n=1
nb−c+1O
µ
1 n
¶
anzn−1
!
and so by (2.10) (1− |z|2)
µ
log 2 1− |z|2
¶α
|(Φb,cf)0(z)| ≤ |Γ(c)| |Γ(b)|
µ
kgklog,α+N(1− |z|2)
µ
log 2 1− |z|2
¶α¶
≤ |Γ(c)|
|Γ(b)|(kgklog,α+N(log 2) α)
<∞, (z∈D).
ThereforeΦb,cf ∈ Blog,α.
Conversely, supposeΦb,cf ∈ Blog,α. From (2.9), g0(z) = Γ(b)
Γ(c)(Φb,cf)
0(z)−X∞
n=1
nb−c+1O
µ
1 n
¶
anzn−1, (z∈D),
and therefore we obtain
kgklog,α≤ ||Γ(c)Γ(b)||kΦb,cfklog,α+ sup z∈D
(1− |z|2)
µ
log 2 1− |z|2
¶α
N
≤ |Γ(b)|
|Γ(c)|kΦb,cfklog,α+N(log 2) α.
Sog∈ Blog,αand the proof is completed. 2
By puttingb = 1andc = 1 +λin the Theorem 2.3 and using (1.4) we obtain the following corollary:
Corollary 2.4 — Let0≤α <ln 2andλ >1. Also letf(z) =Pn∞=1anzn∈ Awhere{an}is a bounded sequence inC. ThenP∞n=1n−λanzn∈ Blog,αif and only ifz−λD−zλf ∈ Blog,α.
Also by puttingb=βandc=γ+βin the Theorem 2.3 and using (1.6) we obtain:
Corollary 2.5 — Let0≤α <ln 2andγ >1. Also letf(z) =Pn∞=1anzn∈ Awhere{an}is a bounded sequence inC. ThenP∞n=1n−γanzn∈ Blog,αif and only if=γβf ∈ Blog,α.
Corollary 2.6 — Suppose0 < b < c−1and f(z) = P∞n=1anzn ∈ H(D)where {an} is a bounded sequence inC. ThenΦb,cfhas nontangential limits in almost every direction.
PROOF: Setg(z) =P∞n=1 an
nc−bzn,(z∈D).By relation (2.9) from Theorem 2.3, we have
Γ(b)
Γ(c)Φb,cf(z) =g(z) +
∞ X
n=1
nb−cO
µ
1 n
¶
anzn, (z∈D).
Sincean=O(1)andc−b >1, we obtain |g(z)| ≤
µ
max n∈N|an|
¶X∞
n=1
1
and
¯ ¯ ¯ ¯ ¯
∞ X
n=1
nb−cO
µ
1 n
¶
anzn
¯ ¯ ¯ ¯ ¯≤
µ
max n∈N|an|
¶X∞
n=1
¯ ¯ ¯ ¯O
µ
1 nc−b+1
¶¯¯ ¯ ¯
<∞, (z∈D).
ThereforeΦb,cf is bounded and so has nontangential limits in almost every direction. 2 The next theorem deals with the case0 < b < c+ 1. For its proof we shall need the following result due to Clunie and Macgrogor [2].
Theorem 2.4 — Letf be analytic and univalent inDand letγ > 1/2. Then there is a setE of measure2πsuch that for allθ∈E, if∠αis a Stolz angle with vertexα=eiθ,
lim z→α
log|f0(z)| ³
log 1 1−|z|
´γ = 0, (z∈∠α).
Theorem 2.5 — Supposef is analytic and univalent inDand0 < b < c+ 1. ThenΦb,cf has
nontangential limits in almost every direction.
PROOF: Letf(z) =P∞n=0anznbe analytic and univalent inDand fixγ in the interval(1/2,1). By Theorem 2.4, there exists some constantAand a setEof measure2πsuch that for allθ∈E
log|f0(z)| ≤A
µ
log 1 1− |z|
¶γ
wheneverα=eiθandz∈∠α. For anyδ >0we have (1− |z|)δ|f0(z)| ≤(1− |z|)δexp
½
A
µ
log 1 1− |z|
¶γ¾
= exp
½
A
µ
log 1 1− |z|
¶γ
−δlog 1 1− |z|
¾
.
Sinceγ <1, it is clear that this last expression approaches zero as|z| →1. Hence |f0(z)| ≤ B
(1− |z|)δ (2.11)
for some constantBwheneverzlies in the Stolz angle∠α. Ifg(z) =
P∞
n=0(c+n)anzn=cf(z) + zf0(z), then from (2.11) we obtain
|g(z)| ≤ |cf(0)|+|c|rB
Z 1
0
1
(1−tr)δdt+r B (1−r)δ
≤ |cf(0)|+ B|c| 1−δ +
rB
for allz∈∠α(|z|=r). Using the same argument as in the proof of Theorem 2.2 we know that Φd,cf(z) = Γ(d)Γ(cΓ(c)−d+ 1)
Z 1
0
td−1(1−t)c−dg(tz)dt. (2.13)
Letdbe such thatb < d < c+ 1, and chooseδ >0so thatδ < c−d+ 1. Then in view of (2.12) and (2.13), for allz∈∠α(|z|=r), we have
|Φd,cf(z)| ≤ Γ(d)Γ(c|Γ(c)−|d+ 1)
Z 1
0
td−1(1−t)c−d|g(tz)|dt
≤ |Γ(c)| Γ(d)Γ(c−d+ 1)
Z 1
0
td−1(1−t)c−d
µ
|cf(0)|+ B|c| 1−δ +
Btr (1−tr)δ
¶
dt
≤ |Γ(c)| Γ(d)Γ(c−d+ 1)
µµ
|cf(0)|+ B|c| 1−δ
¶ Z 1
0
td−1(1−t)c−ddt+B
Z 1
0
td(1−t)c−d−δdt
¶
=K <∞.
HenceΦd,cf is bounded inside∠α. Also Φb,cf(z) = Γ(c)Γ(b)
∞ X
n=0
Γ(b+n) Γ(d+n).
Γ(d+n) Γ(c+n)anz
n
= Γ(c) Γ(b)Γ(d−b)
Z 1
0
tb−1(1−t)d−b−1Ψ(tz)dt (2.14) where
Ψ(z) =
∞ X
n=0
Γ(d+n) Γ(c+n)anz
n, (z∈D).
Therefore
|Ψ(z)|= Γ(d)
|Γ(c)||Φd,cf(z)|< Γ(d)
|Γ(c)|K. (2.15)
Let
h(t) = Γ(c) Γ(b)Γ(d−b)t
b−1(1−t)d−b−1.
Thenh >0,R01h(t)dt= Γ(Γ(dc)), and by (2.14), Φb,cf(z) =
Z 1
0
h(t)Ψ(tz)dt. (2.16)
These properties imply thatΦb,cf is uniformly continuous on the set∠α. To see this, letz1, z2 ∈
∠αand let² >0be given. SinceR01h(t)dtexists, there is anx∈(0,1)such that
Z 1
1−x
h(t)dt < ² 4K
|Γ(c)|
By the uniform continuity ofΨon compact subset ofD, there existsλ >0such that
|Ψ(tz2)−Ψ(tz1)|< 2²Γ(d)|Γ(c)| (2.18)
whenever|z2−z1|< λwithz1, z2 ∈∠αand for allt∈[0,1−x].Therefore, by relations (2.15) to (2.18), for|z2−z1|< λwe have
|Φb,cf(z2)−Φb,cf(z1)| ≤
Z 1−x
0
h(t)|Ψ(tz2)−Ψ(tz1)|dt+
Z 1
1−x
h(t)|Ψ(tz2)−Ψ(tz1)|dt
< ²Γ(d) 2|Γ(c)|
Z 1−x
0
h(t)dt+ 2K Γ(d) |Γ(c)|
Z 1
1−x h(t)dt
< ²Γ(d) 2|Γ(c)|
Z 1
0
h(t)dt+ ² 2
=².
HenceΦb,cfis uniformly continuous inside∠αand so can be extended continuously to the bound-ary. This implies that
lim
z→αΦb,cf(z) (z∈∠α),
exists for everyθ∈E, whereα=eiθ. SinceEhas measure2π, this proves the theorem. 2 ACKNOWLEDGEMENT
The authors are very much thankful to the referee for his/her useful suggestions and comments. REFERENCES
1. K. R. M. Attele, Toeplitz and Hankel operators on Bergman one spaces, Hokkaido Math. J., 21(2) (1992), 279-293.
2. J. G. Clunie and T. H. MacGregor, Radial growth of the derivative of univalent functions, Comment. Math. Helv., 59 (1984), 362-375.
3. P. Galanopoulos, D. Girela, and R. Hernandez, Univalent functions, VMOA and related spaces, J. Geom. Anal., 21(3) (2011), 665-682.
4. D. Girela, Analytic functions of bounded mean oscillation, In: Aulaskari, R. (ed.) Complex Function Spaces, Mekrijarvi 1997. Univ. Joensuu Dept. Math. Rep. Ser. 4 (2001), 61-170.
5. I. B. Jung, Y. C. Kim, and H. M. Srivastava, The Hardy space of analytic functions associated with certain one-parameter families of integral operators, J. Math. Anal. Appl., 176 (1993), 138-147. 6. Y. C. Kim and T. Sugawa, Growth and coefficient estimates for uniformly locally univalent functions on
7. Y. C. Kim and T. Sugawa, Uniformly locally univalent functions and Hardy spaces, J. Math. Anal. Appl., 353 (2009), 61-67.
8. F. W. J. Olver, Asymplotics and special functions, Academic Press, New York, 1974.
9. H. M. Srivastava and S. Owa, New characterizations of certain starlike and convex generalized hyperge-ometric functions, J. Nat. Acad. Math. India, 3 (1985), 198-202.
10. H. M. Srivastava and S. Owa, A certain one-parameter additive family of operators defined on analytic functions, J. Math. Anal. Appl., 118 (1986), 80-87.
11. H. M. Srivastava, M. Saigo, and S. Owa, A class of distortion theorems involving certain operators of fractional calculus, J. Math. Anal. Appl., 131 (1988), 412-420.
