A Class of Analytic Functions associated with Sine Hyperbolic Functions
S. Sivaprasad Kumar, Muhammad Ghaffar Khan, Bakhtiar Ahmad, Wali Khan Mashwani
aa r X i v : . [ m a t h . C V ] N ov A CLASS OF ANALYTIC FUNCTIONS ASSOCIATED WITHSINE HYPERBOLIC FUNCTIONS
S. SIVAPRASAD KUMAR , ∗ , MUHAMMAD GHAFFAR KHAN , BAKHTIAR AHMAD , WALI KHAN MASHWANI Abstract.
We introduce a class of analytic functions subordinate to the func-tion 1+sinh ( z ) and obtain various necessary and sufficient conditions for func-tions to be in the class. These conditions mainly comprise of the coefficientinequalities involving convolution. Further, we have obtained sharp five initialcoefficients, a conjecture for the general nth coefficient and the third Han-kel determinant bounds for the functions in this class. Also derived certaindifferential subordination implication results involving 1 + sinh ( z ). Introduction and Definitions
Let A be the class of all analytic functions f ( z ) defined in the open unit disc D = { z ∈ C : | z | < } with the power series representation as(1.1) f ( z ) = z + ∞ X n =2 a n z n . Further, S denote the class of functions f ∈ A that are univalent in D . Thefunction f ( z ) is subordinate to f ( z ), symbolically written as f ( z ) ≺ f ( z ), ifthere exists a Schwarz function ω ( z ), | ω ( z ) | ≤ | z | , such that f ( z ) = f ( ω ( z )) , ( z ∈ D ) . Furthermore, if the function f belongs to class S , then we have followingequivalence condition f ( z ) ≺ f ( z ) , ( z ∈ D ) if and only if f ( D ) ⊆ g ( D ) and f (0) = g (0). For function f of the form (1 .
1) and g given by g ( z ) = z + ∞ X n =2 b n z n , the Hadamard product or convolution of f and g is defined by( f ∗ g ) ( z ) = z + ∞ X n =2 a n b n z n . Recall that f ( z ) = f ( z ) ∗ z − z and zf ′ ( z ) = f ( z ) ∗ z (1 − z ) . Let P be the class ofanalytic functions k ( z ) with positive real part in D with the normalization(1.2) k ( z ) = 1 + ∞ X n =1 c n z n . Date : ∗ Corresponding author2010
Mathematics Subject Classification.
Key words and phrases.
Analytic functions, Sine hyperbolic function, Subordination, Convo-lution, Third Hankel determinant.
In 1992, Ma and Minda [18] introduced and studied the following subclass of starlikefunctions in A :(1.3) S ∗ ( h ) = (cid:26) f ∈ A : zf ′ ( z ) f ( z ) ≺ h ( z ) ≺ z − z , z ∈ D (cid:27) , where h has positive real part, h ( D ) symmetric about the real axis with h ′ (0) > h (0) = 1. Now by changing the function on the right hand side of (1 . S , which were introduced and investigatedearlier, for example if we set h ( z ) = (1 + Az ) / (1 + Bz ) , where − ≤ B < A ≤ , weobtain Janowski class S ∗ [ A, B ] , see [9]. If h ( z ) = 1 + sin ( z ), we obtain the class S ∗ sin introduced by Cho et al. [5] and also see [2]. By setting h ( z ) = √ z we getthe class S ∗ L , which was introduced and studied by Sok´o l and Stankiewicz [25] andfurther studied by authors in [26]. By varying h ( z ) , following classes are obtained:(1) If h ( z ) = cosh( z ) , Alotaibi et al. [1] introduced and discussed class S ∗ cosh = S ∗ (cosh ( z )).(2) If h ( z ) = 1 + z + z , the class S ∗ Car = S ∗ (cid:0) z + z (cid:1) associated withcardioid introduced by Sharma et al. [22].(3) If h ( z ) = e z , the class S ∗ e = S ∗ ( e z ) was introduced and studied by Mendi-ratta et al. [17] and further investigated by Shi et al. [24].(4) If h ( z ) = z + √ z , Raina and Sokol et al. [20] introduced and discussedthe class S ∗ ℓ = S ∗ (cid:0) z + √ z (cid:1) .(5) If h ( z ) = e − z , recently the class was introduced and discussed by Goeland Kumar [7].(6) If h ( z ) = 1 + z − z , more recently Wani and Swaminathan introducedthe class [27].Also several subclasses of starlike functions were recently introduced in [3, 4, 6,10, 13] by choosing a particular function h ( z ) such as functions associated withBell numbers, functions associated with shell-like curve connected with Fibonaccinumbers or functions connected with the conic domains.Kumar and Gangania [12] consider the analytic univalent function ψ in D suchthat ψ (0) = 0, ψ ( D ) is starlike with respect to 0 and introduced the following classof analytic functions:(1.4) F ( ψ ) := (cid:26) f ∈ A : zf ′ ( z ) f ( z ) − ≺ ψ ( z ) , ψ (0) = 0 (cid:27) . Note that when 1 + ψ ( z ) (1 + z ) / (1 − z ), then the functions in the class F ( ψ )may not be univalent in D which also implies F ( ψ )
6⊆ S ∗ in general. Thus in case,when the function 1 + ψ := h has positive real part, h ( D ) symmetric about thereal axis with h ′ (0) >
0, then F ( ψ ) reduces to the class S ∗ ( h ). With the conditionthat maximum and minimum of the real part of ψ ( z ) is given by ψ ( ± r ), where r = | z | , they established growth theorem and obtained the sharp upper bound fordistortion theorem for the class F ( ψ ). Hence improved the results which was knownfor 0 ≤ α ≤ − √ ≤ β ≤ / BS ( α ) := (cid:26) f ∈ A : zf ′ ( z ) f ( z ) − ≺ z − αz , α ∈ [0 , (cid:27) , where z/ (1 − αz ) =: ψ ( z ) is an analytic univalent function (known as BoothLemniscate function) and symmetric with respect to the real and imaginary axes CLASS OF ANALYTIC FUNCTIONS 3 and S cs ( β ) := (cid:26) f ∈ A : (cid:18) zf ′ ( z ) f ( z ) − (cid:19) ≺ z (1 − z )(1 + βz ) , β ∈ [0 , (cid:27) , where z (1 − z )(1+ βz ) := ψ ( z ) is univalent, analytic, symmetric about the real-axis andmaps the unit disk D onto the domain bounded by Cissoid of Diocles : CS ( β ) := (cid:26) w = u + iv ∈ C : (cid:18) u − β − (cid:19) ( u + v ) + 2 β (1 + β ) ( β − v = 0 (cid:27) studied in [11], [19] and [16].Motivated from the above, we introduce the subclass G sh of F ( ψ ) connected witha sine hyperbolic function as: G sh := (cid:26) f ∈ A : zf ′ ( z ) f ( z ) − ≺ sinh( z ) (cid:27) . Let φ ( z ) := 1+sinh( z ). Note that φ ( z ) is not a Carath´eodery function as ℜ ( φ ( z )) ≯ ∀ z ∈ D . A function f ∈ G sh if and only if there exists an analytic function q satisfying q ( z ) ≺ φ ( z ) such that f ( z ) = z exp (cid:18)Z z q ( t ) − t dt (cid:19) . Thus choosing q ( z ) = φ ( z ), we have the following function(1.5) f ( z ) = z exp (cid:18)Z z sinh( t ) t dt (cid:19) . As a consequence of [12, Theorem 2.1, Corollary 2.1, 2.2, pg 3], we have the sharpresults for the class G sh : Theorem 1.
Let f ∈ G sh and f be given as in (1.5) . Then for | z | = r , we have (1) ( growth theorem ) − f ( − r ) ≤ | f ( z ) | ≤ f ( r ) . (2) ( covering theorem ) either f is a rotation of f or { w ∈ C : | w | ≤ − f ( − } ⊂ f ( D ) , where − f ( −
1) = lim r → ( − f ( − r )) . (3) ℜ f ( z ) z ≤ f ( r ) r and | f ′ ( z ) | ≤ (1 + sinh r ) f ( r ) r . In this paper, we consider some important properties like convolution problems,necessary and sufficient conditions, coefficient problems, convex combination, upperbounds for coefficients, Fekete-szeg¨o problems and third Hankel determinant for theclass G sh .Let f ∈ A , then qth Hankel determinant of f is defined for q ≥ , and n ≥ H q,n ( f ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a n a n +1 . . . a n + q − a n +1 a n +2 . . . a n + q ... ... . . . ... a n + q − a n + q . . . a n +2 q − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Thus second and third Hankel determinants are respectively: H , ( f ) = a a − a ,H , ( f ) = a (cid:0) a a − a (cid:1) − a ( a − a a ) + a (cid:0) a − a (cid:1) . S.S. KUMAR, M. G. KHAN, B. AHMAD, M. K. MASHWANI Preliminary
The following lemmas are important for proving our results.
Lemma 1. [18] . If k ∈ P and it is of the form (1 . , then for λ ∈ C (2.1) | c n | ≤ for n ≥ , and (2.2) (cid:12)(cid:12) c − λc (cid:12)(cid:12) ≤ { | λ − |} . Lemma 2. [18] . If k ∈ P and is represented by (1 . , then (cid:12)(cid:12) c − νc (cid:12)(cid:12) ≤ − ν + 2 ( ν ≤ , ≤ ν ≤ , ν − ν ≥ . Lemma 3. [14, 15] If k ∈ P be expressed in series expansion (1 . , then c = c + x (cid:0) − c (cid:1) for some x , | x | ≤ and c = c + 2 (cid:0) − c (cid:1) c x − (cid:0) − c (cid:1) c x + 2 (cid:0) − c (cid:1) (cid:16) − | x | (cid:17) z for some z , | z | ≤ . Lemma 4. If k ∈ P be expressed in series expansion (1 . , then (2.3) (cid:12)(cid:12) ac − bc c + dc (cid:12)(cid:12) ≤ | a | + 2 | b − a | + 2 | a − b + d | Lemma 5. [21]
Let m, n, l and r satisfy the inequalities < m < , < r < and r (1 − r ) (cid:16) ( mn − l ) + ( m ( r + m ) − n ) (cid:17) + m (1 − m ) ( n − rm ) ≤ m (1 − m ) r (1 − r ) . If k ∈ P and has power series (1 . then (cid:12)(cid:12)(cid:12)(cid:12) lc + rc + 2 mc c − nc c − c (cid:12)(cid:12)(cid:12)(cid:12) ≤ . Lemma 6. [8]
Let w ( z ) be analytic in D with w (0) = 0 . If | w ( z ) | attains itsmaximum value on the circle | z | = r at a point z = re iθ , for θ ∈ [ − π, π ] , we canwrite that z w ′ ( z ) = mw ( z ) , where m is real and m ≥ . Main Results
We begin with the following result:
Theorem 2.
Let f ∈ A be of the form (1 . . Then f ∈ G sh , if and only if (3.1) 1 z f ( z ) ∗ z − βz (1 − z ) ! = 0 , where β = β θ = 1 + sinh (cid:0) e iθ (cid:1) sinh ( e iθ ) . CLASS OF ANALYTIC FUNCTIONS 5
Proof.
Let f ∈ G sh , if and only if zf ′ ( z ) f ( z ) ≺ z ) . if and only if there exist a Schwartz function s ( z ) such that zf ′ ( z ) f ( z ) = 1 + sinh ( s ( z )) ( z ∈ D ) ⇔ zf ′ ( z ) f ( z ) = 1 + sinh (cid:0) e iθ (cid:1) , ( z ∈ D ; θ ∈ [0 , π )) ⇔ z (cid:0) zf ′ ( z ) − f ( z ) (cid:0) (cid:0) e iθ (cid:1)(cid:1) (cid:1) = 0 ⇔ z f ( z ) ∗ z − βz (1 − z ) ! = 0 , where β is as given above and that completes the proof. (cid:3) Note that the forward part of Theorem 2 also holds for β = 1 . As if f ∈ G sh , then f is analytic in D and thus f ( z ) /z = 0 . Theorem 3.
Let f ∈ A be of the form (1 . . Then necessary and sufficient condi-tion for function f ( z ) belong to class G sh is that (3.2) 1 − ∞ X n =2 n − (cid:0) (cid:0) e iθ (cid:1)(cid:1) sinh ( e iθ ) a n z n − = 0 . Proof.
In the light of above Theorem 2, we show that G sh if and only if0 = 1 z " f ( z ) ∗ z − βz (1 − z ) = 1 z ( zf ′ ( z ) − β ( zf ′ ( z ) − f ( z )))= 1 − ∞ X n =2 (( β − n − β ) a n z n − = 1 − ∞ X n =2 n − (cid:0) (cid:0) e iθ (cid:1)(cid:1) sinh ( e iθ ) a n z n − . Hence the proof completes. (cid:3)
Theorem 4.
Let f ∈ A and satisfies (3.3) ∞ X n =2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n − (cid:0) (cid:0) e iθ (cid:1)(cid:1) sinh ( e iθ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | a n | < , then f ∈ G sh . S.S. KUMAR, M. G. KHAN, B. AHMAD, M. K. MASHWANI
Proof.
To show f ∈ G sh , we need to show (3 . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − ∞ X n =2 (( β − n − β ) a n z n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > − ∞ X n =2 (cid:12)(cid:12) (( β − n − β ) a n z n − (cid:12)(cid:12) = 1 − ∞ X n =2 | (( β − n − β ) | | a n | | z | n − > − ∞ X n =2 | (( β − n − β ) | | a n | = 1 − ∞ X n =2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n − (cid:0) (cid:0) e iθ (cid:1)(cid:1) sinh ( e iθ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | a n | > , so by Theorem 3, f ∈ G sh . (cid:3) Theorem 5.
The class G sh is convex.Proof. Let f i ( z ) = z + ∞ X n =2 a n,i z n , for i = { , } . We have to show that µf ( z ) + (1 − µ ) f ( z ) ∈ G sh . As µf ( z ) + (1 − µ ) f ( z )= z + ∞ X n =2 ( µa n, + (1 − µ ) a n, ) z n Consider ∞ X n =2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n − (cid:0) (cid:0) e iθ (cid:1)(cid:1) sinh ( e iθ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | µa n, + (1 − µ ) a n, |≤ µ ∞ X n =2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n − (cid:0) (cid:0) e iθ (cid:1)(cid:1) sinh ( e iθ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | a n, | + (1 − µ ) ∞ X n =2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n − (cid:0) (cid:0) e iθ (cid:1)(cid:1) sinh ( e iθ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | a n, | < µ + (1 − µ ) = 1 . Thus by virtue of Theorem 4, µf ( z ) + (1 − µ ) f ( z ) ∈ G sh . (cid:3) Theorem 6.
Let f ∈ G sh be of the form (1 . . Then | a | ≤ , | a | ≤ , | a | ≤ , | a | ≤ . These inequalities are sharp respectively for f ( z ) = z exp Z z sinh (cid:0) t n − (cid:1) t dt for n = 2 , , , . CLASS OF ANALYTIC FUNCTIONS 7
Proof.
Since f ∈ G sh , then there exists an analytic function s ( z ) , | s ( z ) | < s (0) = 0 , such that(3.4) zf ′ ( z ) f ( z ) = 1 + sinh ( s ( z )) . Denote Ψ ( s ( z )) = 1 + sinh ( s ( z ))and(3.5) k ( z ) = 1 + c z + c z + · · · = 1 + s ( z )1 − s ( z ) . Obviously, the function k ∈ P and s ( z ) = k ( z ) − k ( z )+1 . This gives1 + sinh (cid:18) k ( z ) − k ( z ) + 1 (cid:19) =1 + 12 c z + (cid:18) c − c (cid:19) z + (cid:18) c − c c + 12 c (cid:19) z + (cid:18) − c + 716 c c − c c − c + 12 c (cid:19) z + + · · · . (3.6)And other side, zf ′ ( z ) f ( z ) = 1 + a z + (cid:0) a − a (cid:1) z + (cid:0) a − a a + a (cid:1) z + (cid:0) a − a − a a + 4 a a − a (cid:1) z + · · · . (3.7)On equating coefficients of (3 .
6) and (3 . a = 12 c , (3.8) a = 14 c , (3.9) a = 1144 c − c c + 16 c , (3.10) a = − c + 5192 c c − c c − c + 18 c . (3.11)Using (2 .
1) with equations (3 .
8) and (3 . , we get | a | ≤ | a | ≤ . Application of Lemma 4 to equation (3 . , we get | a | ≤ . Application of Lemma 5 to equation (3 . , we get | a | ≤ . (cid:3) Conjecture 1.
Let f ∈ G sh be of the form (1 . . Then | a n | ≤ n − for n ≥ . As a consequence of above theorem, we have following remark:
S.S. KUMAR, M. G. KHAN, B. AHMAD, M. K. MASHWANI
Remark 1.
Let f ∈ G sh be of the form (1 . . Then for λ ∈ C (cid:12)(cid:12) a − λa (cid:12)(cid:12) ≤
12 max { , | λ − |} and | a − a a | ≤ . For λ = 1 , the above remark gave: Corollary 1.
Let f ∈ G sh be of the form (1 . . Then (cid:12)(cid:12) a − a (cid:12)(cid:12) ≤ . The result is sharp.
Theorem 7.
Let f ∈ G sh be of the form (1 . . Then (cid:12)(cid:12) a a − a (cid:12)(cid:12) ≤ . The result is sharp.Proof.
Since from (3 . .
9) and (3 . (cid:12)(cid:12) a a − a (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) c − c c + 112 c c − c (cid:12)(cid:12)(cid:12)(cid:12) . Using Lemma 3, also put c = c ∈ [0 , , without loss of generality assume | x | = y ∈ [0 ,
1] and eliminating z using triangular inequality, we get (cid:12)(cid:12) a a − a (cid:12)(cid:12) ≤ ψ ( c, y ) , where ψ ( c, y ) := 1288 (cid:18) c + 6 c y (cid:0) − c (cid:1) + 6 c y (cid:0) − c (cid:1) + 12 c (cid:0) − c (cid:1) (cid:0) − y (cid:1) + 92 y (cid:0) − c (cid:1) (cid:19) . Now differentiating ψ ( c, y ) with respect to y , we have ∂ψ ( c, y ) ∂y = 1288 (cid:16) c (cid:0) − c (cid:1) + 12 c y (cid:0) − c (cid:1) − c (cid:0) − c (cid:1) y + 9 y (cid:0) − c (cid:1) (cid:17) , since ∂ψ ( c, y ) ∂y > , thus ψ ( c, y ) is an increasing function and maximum occur at y = 1 , so ψ ( c,
1) = χ ( c ) = 1288 (cid:18) c + 12 c (cid:0) − c (cid:1) + 92 (cid:0) − c (cid:1) (cid:19) , differentiating χ ( c ) with respect to c, we have χ ′ ( c ) = 1288 (cid:0) c + 24 c (cid:0) − c (cid:1) − c − c (cid:0) − c (cid:1)(cid:1) χ ′′ ( c ) = 1288 (cid:0) − c (cid:1) , hence χ ′′ ( c ) < c = 2 , maxima exists at c = 2 , we have (cid:12)(cid:12) a a − a (cid:12)(cid:12) ≤ . CLASS OF ANALYTIC FUNCTIONS 9
This bound is sharp for function defined as follow: f ( z ) = z exp (cid:18)Z z sinh ( t ) t dt (cid:19) = z + z + 12 z + 29 z + · · · , which concludes the proof. (cid:3) Theorem 8.
Let f ∈ G sh be of the form (1 . . Then | H , ( f ) | ≤ ≃ . . Proof.
Since we know that H , ( f ) = a (cid:0) a a − a (cid:1) − a ( a a − a ) + a (cid:0) a − a (cid:1) , using Theorem 6 and 7 and corollary 1, along with triangular inequality, we get thedesired result. (cid:3) Differential Subordination
Theorem 9.
For − ≤ B < A ≤ and f ∈ A , if the conditions (4.1) | α | ≥ A − B − sin 1 − | B | (1 + sinh 1 + cosh 1) and αzf ′ ( z ) ≺ Az Bz , holds. Then f ( z ) z ≺ z. Proof.
Let us define a function(4.2) p ( z ) = 1 + αzf ′ ( z ) , also we consider(4.3) f ( z ) z = 1 + sinh w ( z ) . Now to prove our result, we only need to prove that w ( z ) is a Schwarz function,that is | w ( z ) | < | z | < . Upon logarithmic differentiation of (4 .
3) and using(4 .
2) we obtain p ( z ) = 1 + αz (1 + sinh w ( z ) + zw ′ ( z ) cosh w ( z ))and so (cid:12)(cid:12)(cid:12)(cid:12) p ( z ) − A − Bp ( z ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) α (1 + sinh w ( z ) + zw ′ ( z ) cosh w ( z )) A − B (1 + α (1 + sinh w ( z ) + zw ′ ( z ) cosh w ( z ))) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) α (1 + sinh w ( z ) + zw ′ ( z ) cosh w ( z ))( A − B ) − αB (1 + sinh w ( z ) + zw ′ ( z ) cosh w ( z )) (cid:12)(cid:12)(cid:12)(cid:12) . Now if w ( z ) attains its maximum value at some z = z and | w ( z ) | = 1 . Thenby Lemma 6 for m ≥ , we have, z w ′ ( z ) = mw ( z ) and w ( z ) = e iθ , for θ ∈ [ − π, π ] , we get (cid:12)(cid:12)(cid:12)(cid:12) p ( z ) − A − Bp ( z ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α (cid:0) e iθ + mw ( z ) cosh e iθ (cid:1) ( A − B ) − αB (1 + sinh e iθ + mw ( z ) cosh e iθ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ | α | (cid:0) m (cid:12)(cid:12) cosh e iθ (cid:12)(cid:12) − (cid:12)(cid:12) sinh e iθ (cid:12)(cid:12)(cid:1) ( A − B ) + | α | | B | (1 + | sinh e iθ | + m | cosh e iθ | ) . Since (cid:12)(cid:12) cosh e iθ (cid:12)(cid:12) = cosh (cos θ ) cos (sin θ ) + sinh (cos θ ) sin (sin θ ) = Ψ ( θ ) (cid:12)(cid:12) sinh e iθ (cid:12)(cid:12) = sinh (cos θ ) cos (sin θ ) + cosh (cos θ ) sin (sin θ ) = Θ ( θ ) , one can see that, if we let Ψ ′ ( θ ) = 0 and Θ ′ ( θ ) = 0 has the roots θ = 0 , ± π, ± π in[ − π, π ] , also Ψ ( θ ) and Θ ( θ ) are even functions in this interval somax { Ψ ( θ ) } = Ψ (0) = Ψ ( π ) = cosh (1) , min { Ψ ( θ ) } = Ψ (cid:16) π (cid:17) = cos (1) , max { Θ ( θ ) } = Θ (0) = Θ ( π ) = sinh (1) , min { Θ ( θ ) } = Θ (cid:16) π (cid:17) = sin (1) . From these, we obtain(4.4) cos (1) ≤ (cid:12)(cid:12) cosh e iθ (cid:12)(cid:12) ≤ cosh (1) , (4.5) sin (1) ≤ (cid:12)(cid:12) sinh e iθ (cid:12)(cid:12) ≤ sinh (1) . Now we use (4 .
4) and (4 .
5) to obtain (cid:12)(cid:12)(cid:12)(cid:12) p ( z ) − A − Bp ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≥ | α | (1 + m cos (1) − sin (1))( A − B ) + | α | | B | (1 + sinh (1) + m cosh (1)) . Let φ ( m ) = | α | (1 + m cos (1) − sin (1))( A − B ) + | α | | B | (1 + sinh (1) + m cosh (1)) , which implies φ ′ ( m ) = | α | cos (1) ( A − B ) + | α | B (cos (1) (1 + sinh (1)) − cosh (1) (1 − sin (1)))(( A − B ) + | α | | B | (1 + sinh (1) + m cosh (1))) > , which shows that φ ( m ) is an increasing function and hence it will have its minimumvalue at m = 1 and so (cid:12)(cid:12)(cid:12)(cid:12) p ( z ) − A − Bp ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≥ | α | (1 + cos (1) − sin (1))( A − B ) + | α | | B | (1 + sinh (1) + cosh (1))now by(4 . , we have (cid:12)(cid:12)(cid:12)(cid:12) p ( z ) − A − Bp ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≥ , which contradicts the fact that p ( z ) ≺ (1 + Az ) / (1 + Bz ) and hence we get thedesired result. (cid:3) CLASS OF ANALYTIC FUNCTIONS 11
Corollary 2.
For − ≤ B < A ≤ , and g ∈ A , then if the following conditions | α | ≥ A − B − sin 1 − | B | (1 + sinh 1 + cosh 1) and (4.6) 1 + αz g ′ ( z ) g ( z ) (cid:18) zg ′′ ( z ) g ′ ( z ) − zg ′ ( z ) g ( z ) (cid:19) ≺ Az Bz , holds, then g ∈ G sh . Proof.
Take l ( z ) = z g ′ ( z ) g ( z ) , then we have zl ′ ( z ) = z g ′ ( z ) g ( z ) (cid:18) zg ′′ ( z ) g ′ ( z ) − zg ′ ( z ) g ( z ) (cid:19) and so by (4 . , we get 1 + αzl ′ ( z ) ≺ Az Bz and hence by Theorem 9, we get l ( z ) z = zg ′ ( z ) g ( z ) ≺ z, thus g ∈ G sh . (cid:3) Theorem 10.
For − ≤ B < A ≤ and f ∈ A , then if the conditions (4.7) | α | ≥ ( A − B ) (1 + sinh 1)1 + cos 1 − sin 1 − B (1 + cosh 1 + sinh 1) and (4.8) 1 + α zf ′ ( z ) f ( z ) ≺ Az Bz , holds, then f ( z ) z ≺ z. Proof.
Let p ( z ) = 1 + α zf ′ ( z ) f ( z ) . and f ( z ) z = 1 + sinh w ( z ) , then we have to show that | w ( z ) | < | z | <
1. Now using simple calculations,we obtain that p ( z ) = 1 + α zw ′ ( z ) cosh w ( z ) + sinh w ( z )1 + sinh w ( z )and so (cid:12)(cid:12)(cid:12)(cid:12) p ( z ) − A − Bp ( z ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) α (1 + zw ′ ( z ) cosh w ( z ) + sinh w ( z ))( A − B ) (1 + sinh w ( z )) − Bα (1 + zw ′ ( z ) cosh w ( z ) + sinh w ( z )) (cid:12)(cid:12)(cid:12)(cid:12) . On the contrary if w ( z ) attains its maximum value at some z = z and | w ( z ) | = 1 . Then by Lemma 6 for m ≥
1, we have , z w ′ ( z ) = mw ( z ) , so we have (cid:12)(cid:12)(cid:12)(cid:12) p ( z ) − A − Bp ( z ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α (cid:0) mw ( z ) cosh e iθ + sinh e iθ (cid:1) ( A − B ) (1 + sinh e iθ ) − Bα (1 + mw ( z ) cosh e iθ + sinh e iθ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ | α | (cid:0) m (cid:12)(cid:12) cosh e iθ (cid:12)(cid:12) − (cid:12)(cid:12) sinh e iθ (cid:12)(cid:12)(cid:1) ( A − B ) (1 + | sinh e iθ | ) + | B | | α | (1 + m | cosh e iθ | + | sinh e iθ | ) . Now with the help of (4 .
4) and (4 . (cid:12)(cid:12)(cid:12)(cid:12) p ( z ) − A − Bp ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≥ | α | (1 + m cos (1) − sin (1))( A − B ) (1 + sinh (1)) + | B | | α | (1 + m cosh (1) + sinh (1)) . Now if φ ( m ) = | α | (1 + m cos (1) − sin (1))( A − B ) (1 + sinh (1)) + | B | | α | (1 + m cosh (1) + sinh (1))then φ ′ ( m ) = 2 ( A − B ) | α | cos 1 (sinh 1 + 1) + | B || α | (cos 1 (1 + sinh 1) + cosh 1 (sin 1 − A − B ) (1 + sinh (1)) + | B | | α | (1 + m cosh (1) + sinh (1))) > , which shows that φ ( m ) is an increasing function and hence it will have its minimumvalue at m = 1 and so (cid:12)(cid:12)(cid:12)(cid:12) p ( z ) − A − Bp ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≥ | α | (1 + cos (1) − sin (1))( A − B ) (1 + sinh (1)) + B | α | (1 + cosh (1) + sinh (1))now by(4 . , we have (cid:12)(cid:12)(cid:12)(cid:12) p ( z ) − A − Bp ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≥ , which contradicts (4 .
8) and so | w ( z ) | < | z | <
1, which completes the proof. (cid:3)
We obtain the following result using Theorem 10
Corollary 3.
For − ≤ B < A ≤ , and f ∈ A p then if the condition | α | ≥ ( A − B ) (1 + sinh 1)1 + cos 1 − sin 1 − B (1 + cosh 1 + sinh 1) , and α (cid:18) zg ′′ ( z ) g ′ ( z ) − zg ′ ( z ) g ( z ) (cid:19) ≺ Az Bz , holds then g ∈ G sh . Theorem 11.
For − ≤ B < A ≤ and f ∈ M p then if the condition (4.9) | α | ≥ ( A − B ) (1 + sinh (1)) − sin (1) − B (1 + cosh (1) + sinh (1)) is true and (4.10) 1 + α z f ′ ( z )( f ( z )) ≺ Az Bz , then f ( z ) z ≺ √ z. CLASS OF ANALYTIC FUNCTIONS 13
Proof.
Let p ( z ) = 1 + α z f ′ ( z )( f ( z )) and f ( z ) z = 1 + sinh w ( z ) , then it is to show that | w | < | z | <
1. Also by simplification, we have p ( z ) = 1 + α zw ′ ( z ) cosh w ( z ) + sinh w ( z )(1 + sinh w ( z )) and so (cid:12)(cid:12)(cid:12)(cid:12) p ( z ) − A − Bp ( z ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α (1 + zw ′ ( z ) cosh w ( z ) + sinh w ( z ))( A − B ) (1 + sinh w ( z )) − Bα (1 + zw ′ ( z ) cosh w ( z ) + sinh w ( z )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Now if w ( z ) attains its maximum value at some z = z and | w ( z ) | = 1 . Then byLemma 6 for m ≥ , we have, z w ′ ( z ) = mw ( z ) . So we have (cid:12)(cid:12)(cid:12)(cid:12) p ( z ) − A − Bp ( z ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α (cid:0) mw ( z ) cosh e iθ + sinh e iθ (cid:1) ( A − B ) (1 + sinh e iθ ) − Bα (1 + mw ( z ) cosh e iθ + sinh e iθ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ | α | (cid:0) m (cid:12)(cid:12) cosh e iθ (cid:12)(cid:12) − (cid:12)(cid:12) sinh e iθ (cid:12)(cid:12)(cid:1) ( A − B ) (1 + | sinh e iθ | ) + | B | | α | (1 + m | cosh e iθ | + | sinh e iθ | ) . Now using (4 .
4) and (4 . (cid:12)(cid:12)(cid:12)(cid:12) p ( z ) − A − Bp ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≥ | α | (1 + m cos (1) − sin (1))( A − B ) (1 + sinh (1)) + | B | | α | (1 + m cosh (1) + sinh (1)) . Let φ ( m ) = | α | (1 + m cos (1) − sin (1))( A − B ) (1 + sinh (1)) + | B | | α | (1 + m cosh (1) + sinh (1))which implies φ ′ ( m ) = ( A − B ) | α | cos 1 (1 + sinh (1)) + | B | | α | (cos 1 (1 + sinh 1) + sin 1 cosh 1 − cosh 1) (cid:16) ( A − B ) (1 + sinh (1)) + | B | | α | (1 + m cosh (1) + sinh (1)) (cid:17) > , which shows that φ ( m ) is an increasing function and hence it will have its minimumvalue at m = 1 and so (cid:12)(cid:12)(cid:12)(cid:12) p ( z ) − A − Bp ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≥ | α | (1 + cos (1) − sin (1))( A − B ) (1 + sinh (1)) + | B | | α | (1 + cosh (1) + sinh (1)) . Now by(4 .
9) we have (cid:12)(cid:12)(cid:12)(cid:12) p ( z ) − A − Bp ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≥ , which contradicts (4 . , thus | w ( z ) | < | z | <
1, which yields the desiredresult. (cid:3)
Corollary 4.
For − ≤ B < A ≤ , and g ∈ A p then if the condition | α | ≥ ( A − B ) (1 + sinh (1)) − sin (1) − B (1 + cosh (1) + sinh (1)) , and αg ( z ) zg ′ ( z ) (cid:18) zg ′′ ( z ) g ′ ( z ) − zg ′ ( z ) g ( z ) (cid:19) ≺ Az Bz , holds then g ∈ G sh . Theorem 12.
For − ≤ B < A ≤ and f ∈ M p then if the condition (4.11) | α | ≥ ( A − B ) (1 + sinh (1)) − sin (1) − B (1 + cosh (1) + sinh (1)) , holds and (4.12) 1 + α z f ′ ( z )( f ( z )) ≺ Az Bz , then f ( z ) z ≺ w ( z ) . Proof.
Let us assume p ( z ) = 1 + α z f ′ ( z )( f ( z )) and f ( z ) z = 1 + sinh w ( z ) , then, we need to show that | w | < | z | <
1. Also by rearrangement, we get p ( z ) = 1 + α zw ′ ( z ) cosh w ( z ) + sinh w ( z )(1 + sinh w ( z )) and so (cid:12)(cid:12)(cid:12)(cid:12) p ( z ) − A − Bp ( z ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α (1 + zw ′ ( z ) cosh w ( z ) + sinh w ( z ))( A − B ) (1 + sinh w ( z )) − Bα (1 + zw ′ ( z ) cosh w ( z ) + sinh w ( z )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Now if w ( z ) attains its maximum value at some z = z and | w ( z ) | = 1 . Then byLemma 6 for m ≥ , we have , z w ′ ( z ) = mw ( z ) . So we have (cid:12)(cid:12)(cid:12)(cid:12) p ( z ) − A − Bp ( z ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α (cid:0) mw ( z ) cosh e iθ + sinh e iθ (cid:1) ( A − B ) (1 + sinh e iθ ) − Bα (1 + mw ( z ) cosh e iθ + sinh e iθ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ | α | (cid:0) m (cid:12)(cid:12) cosh e iθ (cid:12)(cid:12) − (cid:12)(cid:12) sinh e iθ (cid:12)(cid:12)(cid:1) ( A − B ) (1 + | sinh e iθ | ) + | B | | α | (1 + m | cosh e iθ | + | sinh e iθ | ) . Now with the help of (4 .
4) and (4 . (cid:12)(cid:12)(cid:12)(cid:12) p ( z ) − A − Bp ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≥ | α | (1 + m cos (1) − sin (1))( A − B ) (1 + sinh (1)) + | B | | α | (1 + m cosh (1) + sinh (1)) . Suppose φ ( m ) = | α | (1 + m cos (1) − sin (1))( A − B ) (1 + sinh (1)) + | B | | α | (1 + m cosh (1) + sinh (1)) , CLASS OF ANALYTIC FUNCTIONS 15 that implies φ ′ ( m ) = ( A − B ) | α | cos 1 (1 + sinh (1)) + | B | | α | (cos 1 + cos 1 sinh 1 + sin 1 cosh 1 − cosh 1) (cid:16) ( A − B ) (1 + sinh (1)) + | B | | α | (1 + m cosh (1) + sinh (1)) (cid:17) > , which shows that φ ( m ) is an increasing function and hence it will have its minimumvalue at m = 1, thus (cid:12)(cid:12)(cid:12)(cid:12) p ( z ) − A − Bp ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≥ | α | (1 + cos (1) − sin (1))( A − B ) (1 + sinh (1)) + | B | | α | (1 + cosh (1) + sinh (1)) . Now by(4 .
11) we have (cid:12)(cid:12)(cid:12)(cid:12) p ( z ) − A − Bp ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≥ , which contradicts (4 . , therefore | w ( z ) | < | z | <
1, which gives the desiredresult. (cid:3)
Corollary 5.
For − ≤ B < A ≤ , and g ∈ A p , then if the condition | α | ≥ ( A − B ) (1 + sinh (1)) − sin (1) − B (1 + cosh (1) + sinh (1)) and α g ( z ) z ( g ′ ( z )) (cid:18) zg ′′ ( z ) g ′ ( z ) − zg ′ ( z ) g ( z ) (cid:19) ≺ Az Bz , holds, then g ∈ G sh . Funding
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All authors jointly worked on the results and they read and approved the finalmanuscript.
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Zesz. Nauk. Politech. Rzeszowskiej Mat. , , SL ∗ . Applied Mathematics and Computation. , , , arXiv.1912.05767v1. , ∗ Department of Applied Mathematics, Delhi College of Engineering Bawana Road,Badli Delhi-110042, India
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