Volume 2010, Article ID 385728,14pages doi:10.1155/2010/385728
Research Article
Convolution Properties of Classes of Analytic and
Meromorphic Functions
Rosihan M. Ali,
1Moradi Nargesi Mahnaz,
1V. Ravichandran,
2and K. G. Subramanian
11School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM Penang, Malaysia
2Department of Mathematics, University of Delhi, Delhi 110 007, India
Correspondence should be addressed to Rosihan M. Ali,[email protected]
Received 30 October 2009; Accepted 13 May 2010
Academic Editor: Sin E. Takahasi
Copyrightq2010 Rosihan M. Ali et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
General classes of analytic functions defined by convolution with a fixed analytic function are introduced. Convolution properties of these classes which include the classical classes of starlike, convex, close-to-convex, and quasiconvex analytic functions are investigated. These classes are shown to be closed under convolution with prestarlike functions and the Bernardi-Libera integral operator. Similar results are also obtained for the classes consisting of meromorphic functions in the punctured unit disk.
1. Motivation and Definitions
LetHUbe the set of all analytic functions defined in the unit diskU:{z:|z|<1}. Denote byAthe class of normalized analytic functionsfz zn∞2anzndefined inU. For two functionsfandginA, the convolution or Hadamard product offandgis the functionf∗g
defined byf∗gz zn∞2anbnzn. A functionf is subordinate to an analytic function
g, writtenfz≺gz, if there exists a Schwarz functionw, analytic inUwithw0 0 and
|wz|<1 satisfyingfz gwz.If the functiongis univalent inU, thenfz≺gzis equivalent tof0 g0andfU⊂gU.
The classes of starlike and convex analytic functions and other related subclasses of analytic functions can be put in the form
S∗g, h:
f∈ A | z
f∗gz
f∗gz ≺hz
whereg is a fixed function andhis a suitably normalized function with positive real part. In particular, let S∗h : S∗z/1−z, hand Kh : S∗z/1−z2, h. For hz 1
1−2αz/1−z, 0≤α < 1,S∗handKhare, respectively, the familiar classesS∗αof starlike functions of orderαandKαconsisting of convex functions of orderα. Analogous to the classS∗g, h, the classKg, his defined by
Kg, h:
f∈ A |1z
f∗gz
f∗gz ≺hz
. 1.2
Letfandg∈ Asatisfy
Re zf
z
fz gz >0, Re
zgz
fz gz >0. 1.3
By adding the two inequalities, it is evident that the functionfz gz/2 is starlike and hence both f and g are close-to-convex and univalent. This motivates us to consider the following classes of functions.
It is assumed in the sequel thatm≥1 is a fixed integer,gis a fixed function inA, and
his a convex univalent function with positive real part inUsatisfyingh0 1.
Definition 1.1. The classSm∗hconsists off:f1, f2, . . . , fm,fk ∈ A,1 ≤k ≤m,satisfying m
j1fjz/z /0 inUand the subordination
mzfkz
m j1fjz
≺hz, k1, . . . , m. 1.4
The classSm∗g, hconsists offfor whichf∗g : f1∗g, f2∗g, . . . , fm∗g ∈ S∗mh. The class Kmhconsists of ffor which zf ∈ Sm∗h, wheref : f1, f2, . . . , fm and zf :
zf1, zf2, . . . , zfm. Equivalently,f ∈ Kmh if fsatisfies the condition m
j1fjz/0 inU and the subordination
mzfkz
m j1fjz
≺hz, k1, . . . , m. 1.5
The classKmg, hconsists offfor whichf∗g∈ Kmh.
Now letf∈ Sm∗handFz mj1fjz/m. From1.4, it follows that
zfkz
The convexity ofhUimplies that
1
m
zmk1fkz
Fz ∈hU, 1.7
which shows that the function F is starlike in U. Thus, it follows from 1.4 that the component function fk of f is close-to-convex in U, and hence univalent. Similarly, the component functionfkoff∈ Kmhis univalent.
Ifm1, then the classesSm∗g, handKmg, hare reduced, respectively, toS∗g, h andKg, hintroduced and investigated in1; these classes were denoted there by Sgh andKgh, respectively. Ifgka, where
kaz: z
1−za, a >0, 1.8
then the classS∗mg, hcoincides with the class studied in2, which there was denoted by
Sah, andKmg, hreduces to a class introduced in3which there was denoted byKah. It is evident that the classesSm∗g, handKmg, hextend the classical classes of starlike and convex functions, respectively.
Definition 1.2. The classCmhconsists off:f1, f2, . . . , fm,fk∈ A,1 ≤k ≤m, satisfying the subordination
mzfkz
m
j1ψjz≺
hz, k1, . . . , m, 1.9
for someψ ∈ Sm∗h. In this case, we say thatf∈ Cmhwith respect toψ ∈ Sm∗h. The class
Cmg, hconsists offfor whichf∗g :f1∗g, f2∗g, . . . , fm∗g ∈ Cmh. The classQmh consists offfor whichzf∈ Cmhor equivalently satisfying the subordination
mzfkz
m j1ϕjz
≺hz, k1, . . . , m, 1.10
for someϕ∈ Kmhwithzϕψ,ψ∈ Sm∗h. In this case, we say thatf∈ Qmhwith respect toϕ∈ Kmh. The classQmg, hconsists offfor whichf∗g∈ Qmh.
Whenm1, the classesCmg, handQmg, hreduce, respectively, toCghandQgh introduced and investigated in1. If g ka, where ka is defined by1.8, then the class
Forα <1, the classRαof prestarlike functions of orderαis defined by
Rα:
f∈ A |f∗ z
1−z2−2α ∈ S ∗α
, 1.11
whileR1consists off∈ Asatisfying Refz/z >1/2.
The well-known result that the classes of starlike functions of order α and convex functions of order α are closed under convolution with prestarlike functions of order α
follows from the following.
Theorem 1.3see4, Theorem 2.4. Letα≤1,φ∈ Rα, andf∈ S∗α. Then
φ∗Hf
φ∗f U⊂coHU, 1.12
for any analytic functionH∈ HU, wherecoHUdenotes the closed convex hull ofHU.
In the following section, by using the methods of convex hull and differential subordination, convolution properties of functions belonging to the four classes Sm∗g, h,
Kmg, h,Cmg, handQmg, h, are investigated. It would be evident that various earlier works, see, for example,5–10, are special instances of our work.
In Section 3, new subclasses of meromorphic functions are introduced. These subclasses extend the classical subclasses of meromorphic starlike, convex, close-to-convex, and quasiconvex functions. Convolution properties of these newly defined subclasses will be investigated. Simple consequences of the results obtained will include the work of Bharati and Rajagopal6involving the functionkaz:1/z1−za,a >0, as well as the work of Al-Oboudi and Al-Zkeri5on the modified Salagean operator.
2. Convolution of Analytic Functions
Our first result shows that the classesS∗mg, hand Kmg, hare closed under convolution with prestarlike functions.
Theorem 2.1. Letm≥1be a fixed integer andga fixed function inA. Lethbe a convex univalent function satisfyingRehz> α,0≤α <1, andφ∈ Rα.
1Iff∈ S∗mg, h, thenf∗φ∈ Sm∗g, h.
2Iff∈ Kmg, h, thenf∗φ∈ Kmg, h.
Proof. 1 It is sufficient to prove that f∗φ ∈ Sm∗h whenever f ∈ Sm∗h. Once this is established, the general result forf∈ Sm∗g, hfollows from the fact that
Fork1,2, . . . , m, define the functionsFandHkby
Fz 1 m
m
j1
fjz, Hkz
zfkz
Fz . 2.2
It will first be proved thatFbelongs toS∗α. Forf∈ S∗mhandz∈ U, clearly
zfkz
Fz ∈hU, k1, . . . , m. 2.3
SincehUis a convex domain, it follows that
1
m
m
k1
zfkz
Fz ∈hU, 2.4
or
zFz
Fz ≺hz. 2.5
Since Rehz> α, the subordination2.5yields
Re zFz
Fz
> α, 2.6
and henceF∈ S∗α.
A computation shows that
zφ∗fk
z
1/mmj1φ∗fj
z
φ∗zfkz
φ∗1/mmj1fj
z
φ∗HkF
z
φ∗Fz . 2.7
Sinceφ∈RαandF∈ S∗α,Theorem 1.3yields
φ∗HkF
z
φ∗Fz ∈coHkU, 2.8
and becauseHkz≺hz, we deduce that
zφ∗fk z
1/mmj1φ∗fj
z ≺hz, k1, . . . , m. 2.9
2The functionfis inKmg, hif and only ifzfis inSm∗g, hand by the first part above, it follows thatφ∗zfzφ∗f∈ S∗mg, h. Henceφ∗f∈ Kmg, h.
Remark 2.2. The above theorem can be expressed in the following equivalent forms:
S∗
m
g, h⊂ Sm∗φ∗g, h, Km
g, h⊂ Km
φ∗g, h. 2.10
When m 1, various known results are easily obtained as special cases of Theorem 2.1. For instance, 1, Theorem 3.3, page 336 is easily deduced fromTheorem 2.11, while 1, Corollary 3.1, page 336follows fromTheorem 2.12. Ifgz kais defined by1.8, then3, Theorem 4, page 110follows fromTheorem 2.11, and3, Corollary 4.1, page 111follows fromTheorem 2.12.
Corollary 2.3. Letm≥1be a fixed integer andga fixed function inA. Lethbe a convex univalent function satisfyingRehz> α,0≤α <1. Define
Fkz
γ1
zγ z
0
tγ−1fktdt
γ∈C, Reγ≥0, k1, . . . , m. 2.11
Iff∈ Sm∗g, h, thenFF1, . . . , Fm ∈ Sm∗g, h. Similarly, iff∈ Kmg, h, thenF∈ Kmg, h.
Proof. Define the functionφby
φz z ∞
n2
γ1
γnz
n. 2.12
For Reγ ≥ 0, the functionφ is a convex function11, and henceφ ∈ Rα 4, Theorem 2.1, page 49. It is clear from the definition ofFkthat
Fkz fkz∗
z ∞
n2
γ1
γnz
n
fk∗φ
z, 2.13
so thatFf∗φ. ByTheorem 2.11, it follows thatFf∗φ∈ Sm∗g, h. The second result is proved in a similar manner.
Remark 2.4. Ifgz kazis defined by1.8, thenCorollary 2.3reduces to2, Theorem 2, page 324.
Theorem 2.5. Letm≥1be a fixed integer andga fixed function inA. Lethbe a convex univalent function satisfyingRehz> α,0≤α <1, andφ∈ Rα.
1If f ∈ Cmg, hwith respect to ψ ∈ Sm∗g, h, thenf∗φ ∈ Cmg, hwith respect to
ψ∗φ∈ Sm∗g, h.
2Iff∈ Qmg, hwith respect toϕ ∈ Kmg, h, then f∗φ ∈ Qmg, hwith respect to
Proof. 1In view of the fact that
f∈ Cm
g, h⇐⇒f∗g ∈ Cmh, 2.14
we well only prove thatf∗φ ∈ Cmhwhenf∈ Cmh. Letf∈ Cmh. Fork 1,2, . . . , m, define the functionsFandHkby
Fz 1 m
m
j1
ψjz, Hkz
zfkz
Fz . 2.15
Sinceψ ∈ S∗mh, it is evident from2.6thatF ∈ S∗α.
Thatψ∗φ∈ S∗mhfollows fromTheorem 2.11. Now a computation shows that
zφ∗fk z
1/mmj1φ∗ψj
z
φ∗zfkz
φ∗1/mmj1ψj
z
φ∗HkF
z
φ∗Fz . 2.16
Sinceφ∈ RαandF∈ S∗α,Theorem 1.3yields
φ∗HkF
z
φ∗Fz ∈coHkU, 2.17
and becauseHkz≺hz, it follows that
zφ∗fk
z
1/mmj1φ∗ψj
z ≺hz, k1, . . . , m. 2.18
Thusf∗φ∈ Cmh.
2The functionfis inQmg, hif and only ifzfis inCmg, hand by the first part, clearlyφ∗zfzφ∗f∈ Cmg, h. Henceφ∗f∈ Qmg, h.
Remark 2.6. Again when m 1, known results are easily obtained as special cases of
Theorem 2.5. For instance,1, Theorem 3.5, page 337follows fromTheorem 2.51, and1, Theorem 3.9, page 339is a special case ofTheorem 2.52.
Corollary 2.7. Letm≥1be a fixed integer andga fixed function inA. Lethbe a convex univalent function satisfyingRehz> α, 0 ≤α < 1. LetFkbe the Bernardi-Libera integral transform offk defined by2.11. Iff∈ Cmg, h, thenFF1, . . . , Fm ∈ Cmg, h.
The proof is similar to the proof ofCorollary 2.3and is therefore omitted.
3. Convolution of Meromorphic Functions
LetΣdenote the class of functionsfof the form
fz 1 z
∞
n0
anzn, 3.1
that are analytic in the punctured unit diskU∗ : {z : 0 < |z| < 1}. The convolution of two meromorphic functionsfandg, wheref is given by3.1andgz 1/z ∞n0bnzn, is given by
f∗gz: 1
z ∞
n0
anbnzn. 3.2
In this section, several subclasses of meromorphic functions in the punctured unit disk are introduced by means of convolution with a given fixed meromorphic function. First we take note that the familiar classes of meromorphic starlike and convex functions and other related subclasses of meromorphic functions can be put in the form
Σsg, h:
f∈Σ| −z
f∗gz
f∗gz ≺hz
, 3.3
wheregis a fixed function inΣandhis a suitably normalized analytic function with positive real part. For instance, the class of meromorphic starlike functions of order α, 0 ≤ α < 1, defined by
Σs:
f ∈Σ| −Rezf
z fz > α
3.4
is a particular case ofΣsg, hwithgz 1/z1−zandhz 1 1−2αz/1−z. Here four classesΣs
mg, h,Σkmg, h,Σcmg, h, andΣ q
mg, hof meromorphic functions are introduced and the convolution properties of these new subclasses are investigated. As before, it is assumed thatm≥1 is a fixed integer,gis a fixed function inΣ, andhis a convex univalent function with positive real part inUsatisfyingh0 1.
Definition 3.1. The classΣs
mhconsists off:f1, f2, . . . , fm,fk ∈Σ,1 ≤k ≤ m,satisfying m
j1fjz/0 inU∗and the subordination
−mzfkz m j1fjz
The classΣs
mg, hconsists offfor whichf∗g : f1∗g, f2 ∗g, . . . , fm∗g ∈ Σsmh. The class Σk
mh consists offfor which −zf ∈ Σsmhor equivalently satisfying the condition m
j1fjz/0 inU∗and the subordination
−m
zfkz
m j1fjz
≺hz, k1, . . . , m. 3.6
The classΣk
mg, hconsists offfor whichf∗g∈Σkmh.
Various subclasses of meromorphic functions investigated in earlier works are special instances of the above defined classes. For instance, ifgz: 1/z1/1−z, thenΣs
mg, h coincides withΣs
mh. By puttinggpμ∗qβ,λ, where
pμz: 1
z1−zμ, qβ,λz:
1
z ∞
k0
λ k1λ
β
zk, 3.7
the classΣs
mg, hreduces to the classΣ β
λ,μm, hinvestigated in9. Ifgkn, where
knz: 1
z ∞
k1
1λk1nzk, 3.8
then the class ofΣs
mg, his the classΣmn, λ, hstudied in5. Ifgka, where
kaz: 1
z1−za, a >0, 3.9
then the classΣs
mg, hcoincides withΣma, hinvestigated in6. Definition 3.2. The classΣc
mhconsists off: f1, f2, . . . , fm,fk ∈ Σ,1 ≤k ≤ m,satisfying the subordination
−mzfkz m j1ψjz
≺hz, k1, . . . , m, 3.10
for someψ ∈Σsmh. In this case, we say thatf∈Σcmhwith respect toψ ∈Σsmh. The class
Σc
mg, hconsists offfor whichf∗g :f1∗g, f2∗g, . . . , fm∗g ∈Σcmh. The classΣ q mh consists offfor which−zf∈Σc
mhor equivalently satisfying the subordination
−m
zfkz
m j1ϕjz
≺hz, k1, . . . , m, 3.11
for someϕ∈ Kmhwith−zϕψandψ ∈ Sm∗h. The classΣ q
mg, hconsists offfor which
Ifgz : 1/z 1/1−z, thenΣc
mg, hcoincides withΣcmh. Ifgz knzis defined by3.8, thenΣc
mg, hreduces toQmn, λ, hinvestigated in5. Ifgz kazis defined by3.9, then the classΣc
mg, his the classC∗ma, hstudied in6.
We shall require the theorem below which is a simple modification ofTheorem 1.3.
Theorem 3.3. Letα ≤ 1,f, φ ∈ Σ,z2φ ∈ R
α, andz2f ∈ S∗α. Then, for any analytic function
H∈ HU,
φ∗Hf
φ∗f U⊂coHU. 3.12
Theorem 3.4. Assume thatm≥1is a fixed integer andgis a fixed function inΣ. Lethbe a convex univalent function satisfyingRehz<2−α,0≤α <1, andφ∈Σwithz2φ∈ Rα.
1Iff∈Σs
mg, h, thenf∗φ∈Σsmg, h.
2Iff∈Σk
mg, h, thenf∗φ∈Σkmg, h.
Proof. 1It is enough to prove the result forgz 1/z1−z.Fork1,2, . . . , m, define the functionsFandHkby
Fz 1 m
m
j1
fjz, Hkz −
zfkz
Fz . 3.13
We show thatFsatisfies the conditionz2F∈ S∗α. Forf∈Σs
mhandz∈ U, clearly
−zfkz
Fz ∈hU, k1, . . . , m. 3.14
SincehUis a convex domain, it follows that
−1
m
m
k1
zfkz
Fz ∈hU, 3.15
or
−zFz
Fz ≺hz. 3.16
Since Rehz<2−α, the subordination3.16yields
−Re zF
z Fz
and thus
Re
zz2Fz
z2Fz
RezF
z
Fz 2> α. 3.18
Inequality3.18shows thatz2F∈ S∗α.
A routine computation now gives
− z
φ∗fk z
1/mmj1φ∗fj
z
φ∗−zfkz
φ∗1/mmj1fj
z
φ∗HkF
z
φ∗Fz . 3.19
Sincez2φ∈R
αandz2F∈ S∗α,Theorem 3.3yields
φ∗HkF
z
φ∗Fz ∈coHkU, 3.20
and becauseHkz≺hz, it is clear that
− z
φ∗fk
z
1/mmj1φ∗fj
z ≺hz, k1, . . . , m. 3.21
Thusf∗φ∈Σs mh.
2The functionfis inΣkmg, hif and only if−zfis inΣsmg, hand the result of part
1shows thatφ∗−zf −zφ∗f∈Σs
mg, h. Henceφ∗f∈Σkmg, h.
Remark 3.5. 1The above theorem can be written in the following equivalent forms:
Σs m
g, h⊂Σsmφ∗g, h, Σkmg, h⊂Σkmφ∗g, h. 3.22
2 When m 1, various known results are easily obtained as special cases of
Theorem 3.4. For instance, ifgz pμ ∗qβ,λ is defined by3.7, then 9, Theorem 6, page 1265follows fromTheorem 3.41.
Corollary 3.6. Assume thatm≥1is a fixed integer andgis a fixed function inΣ. Lethbe a convex univalent function satisfyingRehz<2−α, 0≤α <1. Define
Fkz
γ1
zγ2 z
0
tγ1fktdt
γ∈C, Reγ≥0, k1, . . . , m. 3.23
Iff∈Σs
Proof. Define the functionφby
φz 1 z
∞
n0
γ1
γ2nz
n. 3.24
For Reγ ≥ 0, the function z2φz is a convex function 11, and hence z2φz ∈ R α 4, Theorem 2.1, page 49. It is clear from the definition ofFkthat
Fkz fkz∗
1
z ∞
n0
γ1
γ2nz
n
fk∗φ
z, 3.25
so thatFf∗φ. ByTheorem 3.4, it follows thatFf∗φ∈Σs mg, h. The second result is established analogously.
Remark 3.7. Again we take note of how our results extend various earlier works. Ifgz pμ∗qβ,λ is defined by3.7, then 7, Proposition 2, page 512follows fromCorollary 3.6. If
gz knzis defined by3.8, thenCorollary 3.6yields5, Theorem 2.2, page 4. Ifgz
kazis defined by3.9, thenCorollary 3.6reduces to6, Theorem 2, page 11.
Theorem 3.8. Assume thatm≥1is a fixed integer andgis a fixed function inΣ. Lethbe a convex univalent function satisfyingRehz<2−α,0≤α <1, andφ∈Σwithz2φ∈ R
α.
1If f ∈ Σc
mg, hwith respect to ψ ∈ Σsmg, h, thenf∗φ ∈ Σcmg, hwith respect to
ψ∗φ∈Σs mg, h.
2If f ∈ Σqmg, h with respect toϕ ∈ Σkmg, h, thenf∗φ ∈ Σ q
mg, hwith respect to
ϕ∗φ∈Σk mg, h. Proof. 1In view of the fact that
f∈Σcmg, h⇐⇒f∗g ∈Σcmh, 3.26
it is sufficient to prove thatf∗φ∈Σc
mhwhenf∈Σcmh. Letf∈Σcmh. Fork1,2, . . . , m, define the functionsFandHkby
Fz 1 m
m
j1
ψjz, Hkz −
zfkz
Fz . 3.27
Inequality3.18shows thatz2F∈ S∗α.
It is evident that
− z
φ∗fk z
1/mmj1φ∗ψj
z
φ∗−zfkz
φ∗1/mmj1ψj
z
φ∗HkF
z
Sincez2φ∈ R
αandz2F∈ S∗α,Theorem 3.3yields
φ∗HkF
z
φ∗Fz ∈coHkU, 3.29
and becauseHkz≺hz, it follows that
− z
φ∗fk
z
1/mmj1φ∗ψj
z ≺hz, k1, . . . , m. 3.30
Thusf∗φ∈Σc mh.
2The functionfis inΣqmg, hif and only if−zfis inΣcmg, hand from the first part above, it follows thatφ∗−zf −zφ∗f∈Σc
mg, h. Henceφ∗f∈Σ q mg, h.
Corollary 3.9. Assume thatm≥1is a fixed integer andgis a fixed function inΣ. Lethbe a convex univalent function satisfyingRehz<2−α,0≤α <1. LetFkbe defined by3.23. Iff∈Σcmg, h, thenFF1, . . . , Fm ∈Σcmg, h.
The proof is analogous toCorollary 2.3and is omitted.
Remark 3.10. If gz knz is defined by3.8, then Corollary 3.9yields5, Theorem 3.1, page 9. Ifgz kazis defined by3.9, thenCorollary 3.9reduces to6, Theorem 4, page 14.
Acknowledgment
The work presented here was supported in part by research grants from Universiti Sains Malaysia and University of Delhi.
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