In this paper, we clarify some throughout examples, and we also show the equivalence between the complex version of Rolle''s theorem and Lagrangle’s mean value theorem.
L.T.T.Huyen, L.P.Linh/No… No.24_December 2021 TẠP CHÍ KHOA HỌC ĐẠI HỌC TÂN TRÀO TẠP-CHÍ ISSN: 2354 1431KHOA HỌC ĐẠI HỌC TÂN TRÀO ISSN: 2354 - 1431 http://tckh.daihoctantrao.edu.vn/ http://tckh.daihoctantrao.edu.vn/ THE MEAN VALUE THEOREM FOR HOLOMORPHIC FUNCTIONS OF THE MEAN VALUE THEOREM FOR HOLOMORPHIC FUNCTIONS OF A COMPLEX VARIABLE A COMPLEX VARIABLE Luu Thi Thu Huyen1,*, Luu Phuong Linh2,** Luu Thi Thu Huyen1,*, Luu Phuong Linh2,** 1 Hung Vuong University, Vietnam Hung Vuong University, Vietnam 2 Hung Vuong High Quality High School Hung Vuong High Quality High School * Địa email: luuhuyen87@gmail.com * Địa email: luuhuyen87@gmail.com ** ** Địa email: tunhienvnu@gmail.com Địa email: tunhienvnu@gmail.com https://doi.org/10.51453/2354-1431/2021/549 https://doi.org/10.51453/2354-1431/2021/549 Article info Received: 20/5/2021 Accepted: 1/12/2021 Keywords: Mean value theorem, Mean value theorem in , Complex Rolle’s theorem, Flett’s theorem, Myers’s theorem Abstract The mean value theorem is one of the most fundamental results in real analysis However, it fails for holomorphic function of a complex variable even if the function is differentiable on whole complex plane In this paper, we clarify some throughout examples, and we also show the equivalence between the complex version of Rolle's theorem and Lagrangle’s mean value theorem |181 L.T.T.Huyen, L.P.Linh/No… No.24_December 2021 TẠP CHÍ KHOA HỌC ĐẠI HỌC TÂN TRÀO ISSN: TẠP 2354 CHÍ - 1431 KHOA HỌC ĐẠI HỌC TÂN TRÀO http://tckh.daihoctantrao.edu.vn/ ISSN: 2354 - 1431 http://tckh.daihoctantrao.edu.vn/ ĐỊNH LÝ GIÁ TRỊ TRUNG BÌNH ĐỐI HÀM GIẢI TÍCH ĐỊNH LÝ GIÁ TRỊVỚI TRUNG BÌNH ĐỐI VỚIMỘT HÀMBIẾN GIẢI PHỨC TÍCH MỘT BIẾN PHỨC 2,** 2,** Lưu Thị Thu ThuHuyền Huyền1,*,1,*Lưu , Lưu Phương Lưu Thị Phương LinhLinh 11 Đại học họcHùng HùngVương, Vương, Đại ViệtViệt NamNam 22 TrườngPhổ Phổthông thông Chất lượng Vương Trường Chất lượng cao cao HùngHùng Vương ** Địa email: email:luuhuyen87@gmail.com luuhuyen87@gmail.com Địa ** ** Địa email: tunhienvnu@gmail.com Địa email: tunhienvnu@gmail.com https://doi.org/10.51453/2354-1431/2021/549 https://doi.org/10.51453/2354-1431/2021/549 Thông tin viết Ngày nhận bài: 20/5/2021 Ngày duyệt đăng: 1/12/2021 Từ khoá: Định lý giá trị trung bình, Định lý giá trị trung bình , Định lý Rolle , Định lý Fellt, Định lý Myers Tóm tắt Định lý giá trị trung bình kết giải tích thực Tuy nhiên khơng cịn cho hàm chỉnh hình biến phức, kể trường hợp hàm khả vi toàn mặt phẳng phức, báo làm rõ điều số ví dụ cụ thể, đồng thời tương đương Định lý Rolle Định lý giá trị trung bình mặt phẳng phức Introduction It is known that the mean value theorem is one of the most fundamental results in analysis Some deformations of this theorem were shown, such as Flett’s theorem [1858], Myers’s theorem [1977], Sahoo Riedel’s theorem [1998] These theorems have many applications in analysis, in solving equations, systems of equations, finding solutions or breakpoints of polynomials, used to solve many optimization problems, economy,… [3] 182| Let f be a continuous function on a closed interval a, b The difference between the values of f at the endpoints of a, b , if the derivative f '(a) exists, and one gets: f (b) − f (a) f '(a).(b − a) (1) Set x = b − a , then (1) can be written as: (2) f (a + x) f (a) + f '(a).x Formula (1) can be replaced by: f (b) − f (a= ) f '(c).(b − a) (3) Luu Thi Thu Huyen et al/No.24_Dec 2021|p181-187 L.T.T.Huyen, L.P.Linh/No… where c (a, b) , the function f is differentiable at every point of (a, b) This is c ( a, b ) there is a point such that f '(c) = (5) the result which known as the Mean value theorem In the complex analysis, a natural question arises: Does the mean value theorem still hold in the field of complex numbers? The results of the paper is developped based on the two papers [1], [5] and [7] Its content analyzes carefully some examples showing that, mean value theorem and the some theorems were developed from it fails for complex valued functions, even if the function is differentiable on whole the complex plane At the same time, the equivalence of Rolle’s and Mean value theorems in the complex plane are proved in detail, many calculations are more detailed than the original document Content a) Mean value theorem and some its deformations Theorem (Largrange’s mean value theorem) Rolle’s theorem and Largrange’s mean value theorem in real-valued functions are equivalent [5] Proof of the equivalence i) It is easy, Theorem deduce Theorem Suppose f satisfy the conditions of Let f be real continuous function on a, b g (a) = g (b) Then, by Theorem 2: there and differentiable in ( a, b ) There exists a f (b) − f (a= ) f '(c).(b − a) (4) If f (a) = f (b) , then the mean value theorem reduces to Rolle’s theorem which is also the another most fundamental result in real analysis Theorem (Rolle’s theorem) Let f be a real continuous function on and differentiable Furthermore, assume By f (a) = f (b) so f '(c)(b − a) = It deduce f '(c) = ii) We will prove Theorem deduce Theorem Indeed: Suppose f satisfy the conditions of Theorem Consider the auxiliary function: f ( x) g ( x) = x f (a ) a f (b) b = (a − b) f ( x) − ( x − b) f (a ) + ( x − a ) f (b) We see that g ( x) is a real function: a, b , ( a, b ) and continuous on the closed interval differentiable in the open interval exists a point c ( a, b ) such that g'(c) = More g '( x) = (a − b) f '( x) − f (a) + f (b) point c ( a, b ) such that a, b Theorem Then f (b) − f (a)= f '(c)(b − a) in ( a, b ) Hence, g '(c) = (a − b) f '(c) − f (a) + f (b) = f (b) − f (a)= f '(c)(b − a) Which proves that Theorem deduce Theorem So, two theorems are equivalent Next, we consider some theorems developed from the Mean value theorem Theorem (Flett’s Mean Value Theorem [2]) f (a) = f (b) Then |183 Luu Thi Thu Huyen et al/No.24_Dec 2021|p181-187 L.T.T.Huyen, L.P.Linh/No… be differentiable on a, b f ( z) − f = (0) f '( z)( z −= 0) f '( z) z has no and f '(a) = f '(b) Then there exists a point solution in the interval (0, 2 i) Indeed, the Let f : a, b → c ( a, b ) such that f (c) − f (a)= f '(c).(c− a) (6) Theorem (Myers’s Mean Value Theorem [4]) be differentiable on a, b Let f : a, b → equation f ( z) − = f (0) f '( z)( z − 0) = − z e− z When z = iy , we have: − iy = e−iy = cos y − i sin y , the comparison of the real and imaginary parts gives: cos y = sin y = y and f '(a) = f '(b) Then there exists a point c ( a, b ) such that f (b) − f (c) = f '(c).(b − c) (7) Theorem (Sahoo Riedel’s Theorem [6]) be differentiable on a, b Let f : a, b → Then there exists a point c ( a, b ) such that f (c ) − f ( a ) = f '(c)(c − a) − f '(b) − f '(a) (c − a ) b−a (8) Remark: Myers’s theorem is a complete complement to Flett’s theorem When f '(a) = f '(b) then Sahoo and Riedel’s Evidently, this system of equations has no solution in the interval (0, 2 ) This means f ( z) − f (0)= f '( z)( z − 0) has no solution in the interval (0, 2 i) Thus Flett’s theorem fails in the complex domain For the case of Largrange’s Mean Value Theorem, consider the function z − ( z1 + z2 ) f ( z ) = exp i , z1 , z2 are two z2 − z1 distinct points in the complex plane We can calculate: z − (z1 + z ) −i = e = −1; f (z1 ) = exp i z2 − z1 z − (z1 + z ) i = e = −1; f (z ) = exp i z2 − z1 theorem become Flett’s theorem By analysing some examples in both real and complex analysis, we show that the Mean value theorem and some its deformations z − (z1 + z ) 2i f '(z) exp i = 0, not hold for holomorphic functions of one z2 − z1 z2 − z1 complex variable Such as: z For the case of Flett’s theorem, consider the Therefore f ( z2 ) − f ( z1 ) − f '( z )( z2 − z1 ) function f ( z ) =e z − z , z We see that f =−1 + − f '( z )( z2 − z1 ) is holomorphic and f '( z= ) e z −1 , therefore f '(2k i) = e2k i −1 = k Consider 0, 2 i , we have the closed interval f '(0) = f '(2 i) Nevertheless, 184| =− f '( z )( z2 − z1 ) 0, z So f ( z2 ) − f ( z1= ) f '( z)( z2 − z1 ) has no solution Thus Largrange’s mean value theorem fails in the complex domain Luu Thi Thu Huyen et al/No.24_Dec 2021|p181-187 L.T.T.Huyen, L.P.Linh/No… For the case of Rolle’s theorem, consider the (0) = (b1 − a1 )u (a) + (b2 − a2 )v(a) = 0; (1) = (b1 − a1 )u (b) + (b2 − a2 )v(b) = function f ( z) =e − 1, z We see that z f (2k i)= 0, k z but f '( z= ) e= has no solution in the complex plane Thus Rolle’s theorem fails in the complex domain The previous examples show that the Mean value theorem does not hold in the complex plane if we fix the hypothesis as in the case of real analysis Next, we present the similar theorems which hold in complex analysis b) Rolle’s and Mean value theorem in complex plane [1],[7] Let a and b be distinct points in Denote a, b the set a + t (b − a) : t 0,1 and we will refer to a, b as a line segment or a closed interval in Similarly, (a, b) denotes the set a + t (b − a ) : t (0,1) Theorem (Complex Rolle’s theorem) Let f be a holomorphic function defined on an open convex subset D f of Let a and b be two distinct points in Df satisfy there exist f= (a) f= (b) Then z1 , z2 (a, b) such that Re( f '( z1 )) = and Im( f '( z2 )) = Proof Let According to Rolle’s theorem, it exists t1 (0,1) such that '(t1 ) = Let z1 =+ a t1 (b − a) We have u dx u dy 0= '(t1 ) = (b1 − a1 ) + x dt y dt v dx v dy + (b − a ) + x dt y dt u u = (b1 − a1 ) (z1 ).(b1 − a1 ) + (z1 ).(b − a ) y x v v +(b − a ) (z1 ).(b1 − a1 ) + (z1 ).(b − a ) y x u u (z1 ) + (b1 − a1 ).(b − a ) (z1 ) x y v v +(b − a ).(b1 − a1 ) (z1 ) + (b − a ) (z1 ) x y u = (b1 − a1 ) + (b − a ) ( z1 ) x = (b1 − a1 ) u = ( z1 ) x Apply to the function g = −if , it exists '( z1 )) So Re( f= z2 (a, b) such that: v u 0= Re( g '( z2 )) = ( z2 ) = − ( z2 ) = Im( f '( z2 )) x y Theorem (Complex Mean value theorem) Let f be a holomerphic function defined on an open convex subset D f of a= a1 + ia2 , b = b1 + ib2 , f (= z ) u ( z ) + iv( z ) z D f and b be two distinct points in D f Then there exist z1 , z2 (a, b) such that f (b) − f (a) Re( f '( z1 )) = Re b−a (t ) = (b1 − a1 )u (a + t (b − a)) + (b − a2 )v(a + t (b − a)), t [0,1] By hypothesis f= (a) f= (b) so u= (a) v= (a) u= (b) v= (b) Let a and and f (b) − f (a) Im( f '( z2 )) = Im b−a We can calculate: |185 Luu Thi Thu Huyen et al/No.24_Dec 2021|p181-187 L.T.T.Huyen, L.P.Linh/No… Assume that f satisfy the conditions of Proof Let g ( z ) = f ( z ) − f (a) − z D f f (b) − f (a) ( z − a), b−a It is easy to see g (a) = g (b) Theorem Consider the auxiliary function: g ( z) = a −b By Theorem 6, there exists z1 , z2 (a, b) f ( z ) − f (a) = such that Re( g '( z1 )) = and Im( g '( z2 )) = Furthermore f (b) − f (a) g '(= z ) f '( z ) − , z D f So b−a f (b) − f (a ) = = g '( z1 )) Re( f '( z1 )) − Re Re( b−a f (b) − f (a ) Re( f '( z1 )) = Re ; b−a f ( z) z f (a) a f (b) b z −b z−a + f (b) a −b a −b (9) We see that g ( z ) is a holomorphic function in Df and satisfy the condition g= (a) g= (b) Then, by Theorem 6, exists z1 , z2 (a, b) such that Re( f '( z1 )) = and Im( f '( z2 )) = From (9) we have: f (b) − f (a) g '(= z ) f '( z ) − , z D f , so f (b) − f (a ) b−a = = Im( g '( z2 )) Im( f '( z2 )) − Im b−a f (b) − f (a) = = Re( g '( z1 )) Re( f '( z1 )) − Re f (b) − f (a ) b−a Im( f '( z2 )) = Im b−a f (b) − f (a ) Re( f '( z1 )) = Re ; We see Theorem and Theorem are b−a equivalent Indeed: f (b) − f (a) = = Im( g '( z2 )) Im( f '( z2 )) − Im i) Theorem implies Theorem 6: Assume b−a that f function satisfy the conditions of f (b) − f (a) Im( f '( z2 )) = Im b−a Theorem Then, we have Then, Theorem implies Theorem f (b) − f (a) Re( f '( z1 )) = Re Therefore, Theorem and Theorem are b−a equivalent and Conclusion f (b) − f (a) Im( f '( z2 )) = Im , By studying the results, the authors b−a had to clarify the picture of Mean Value z1 , z2 (a, b) Moreover f= (a) f= (b) Theorem in Complex plane By illustrating in f ( b ) − f ( a ) details some examples, we showed that, Re therefore and =0 b − a Rolle’s and Mean Value Theorem fails for holomorphic functions of a complex variable, f (b) − f (a) Im = , hence Re( f '( z1 )) = even if the function is differentiable b−a throughout the complex plane The proof of and Im( f '( z2 )) = the theorems were presented in details In ii) Theorem implies Theorem particular, the paper proved the equivalent 186| Luu Thi Thu Huyen et al/No.24_Dec 2021|p181-187 L.T.T.Huyen, L.P.Linh/No… between Rolle’s theorem and Largrange’s Mean value theorem, in which the calculation in details added REFERENCES [1] J.C Evard and F Jafari, A complex Rolle’s theorem, Amer Math Monthly, 99 (1992), 858-861 [2] T.M Flett, A mean value theorem, Math Gazette, 42 (1958), 38-39 [3] Y Kaya, Complex Rolle and Mean Value Theorems, MSc Thesis, Balıkesir University, (2015) [4] R.E Myers, Some elementary results related to the mean value theorem, The two- year college Mathematics journal, 8(1) (1977), 51-53 [5] M.A Qazi, The mean value theorem and analytic functions of a complex variable, J Math Anal Appl 324 (2006) 30 – 38 [6] P.K Sahoo and T.R Riedel, Mean value theorem and functional equations, World Scientific, River Edge, New Jersey, 1998 [7] Sỹmeyra Uỗar and Nihal ệzgỹr, Complex conformable Rolle’s and Mean Value Theorems, Mathematical Sciences, Vol 14, no 3, Sept 2020 |187 ... a) Mean value theorem and some its deformations Theorem (Largrange’s mean value theorem) Rolle’s theorem and Largrange’s mean value theorem in real-valued functions are equivalent [5] Proof of. .. related to the mean value theorem, The two- year college Mathematics journal, 8(1) (1977), 51-53 [5] M .A Qazi, The mean value theorem and analytic functions of a complex variable, J Math Anal Appl... whole the complex plane At the same time, the equivalence of Rolle’s and Mean value theorems in the complex plane are proved in detail, many calculations are more detailed than the original document