This is not exactly an answer which describes a single study (but studies are part of what I searched through).
What I did: I searched systematically google using appropriate words "backlit lcd vs e ink epaper eye strain" and looked at all the first results (first google search sites). I extracted all the important text parts of these links and attached these to this post.
My own conclusion is this: Using a lcd display does not make an eye strain difference / is not better than eink/epaper if you're using a ...
- appropriate brightness: Many people use too bright displays. For correct configuration you can do this: Make this monitor as bright as possible and then reduce the brightness and while you do that always switch looks beside the monitor and notice if you percieve the monitor as a light source. If you don't find your monitor a light source, you got the correct brightness (notice: because of ambient light changes [e.g. day, night] your monitor brightess must be adjusted respectively for perfect monnitor brightness). On the other side, when you're using a monitor/ tablet display with high ambient light (e.g. outside) it is important your display supports high enough brightness (e.g. high cd/m²), to also adjust to that ambient light. E.g. 500-600 cd/m² is no problem for new tablet displays (btw. sometimes just rotating your tablet in some direction has the effect, that you don't have to have a very, very high brightness [so e.g. don't hold it directly in the light]).
- appropriate anti-glare: Because many product developers want "brilliant" screens the often use glare displays. The problem with this is, that you will have reflections on that which can cause eye strain. Many try ti circumvent that by using a too high brightness. The in my opion much better way is to use an anti-glare display or if you already got an glare display, you can buy a screen display protector film which is anti glare. (btw. the kindle paperwhite also uses an anti glare display)
- continous backlight (no flicker): Some monitors (and also some tablet displays) adjust the brightness using PWM. This means to give the impression of lower light they don't just turn the light volume lower but the use full light volume and then just turn the screen black for a short time period. This isn't good for the eyes and can also be audible on low light cause of a high pitched tone. So before buying a monitor / tablet, make sure that it uses continious backlight. Btw. LCD pixel itself do not flicker or something if you have still image. They just make the transition from one state to the other (and with too much "overdrive" you can have short white flashes while transitioning [short higher voltage for faster transition, configureable in many monitores], so turn off overdrive or use it small). But when you have a still image (while reading), there is nothing to change and the still image will just continously be displays (no flicker or something).
- appropriate light color: When you are reading a real book / eink (turned off light) you don't have as much blue light as often people have on an backlit lcd screen. You can adjust that on your lcd using e.g. f.lux (windows) or twilight (android - but better are apps using root rights cause dark colors then can be changed more appropriately).
- appropriate resolution: On an backlit lcd screen you have pixels which are often blurred/grainy cause of to small resolution. E.g. on a 27 inch display with fullHD you get arount 80 ppi. A kindle paperwhite has 300 ppi. So get an 4k display, if you can: This will get you around 160 ppi which is still less, but keep in mind that your distance is also doubled in comparison to the distance you have to a kindle paperwhite while reading (so this is pretty nice resolution), Also when you buy an tablet you can easily get tablets with around 250 ppi.
- appropriate angle independence: There are some monitors which are highly dependent on view angle. So it can happen that your two eyes don't see just two differnece perspectives but also difference colors and so on. But at todays monitors with even TN-Panels it should be no problem. And on IPS-Panels that's no problem anyway.
Here an intuitive metapher why lcd displays fullfilling the above points do not differ from eink in terms of eye strain: When your eink device gets sunlight on it, it reflects them to your eye. But suppose you could build a display (we call now "D.") that you emit exactly the same light that yould normally reflect your eink device. Light is light, so that would be exactly the same for you, you couldn't say that it is reflected or emitted light. But now look on your lcd display fullfilling the above points (we call it "L."). Oh, L. already equals D. (fullfilling above points), so it also equals the eink display :).
Extracted google search
- https://www.howtogeek.com/181577/e-ink-vs.-lcd-which-screen-is-best-for-reading/ (refering to a study)
- "The key here is that the LCD screen has to be a high resolution, which modern tablet LCD screens are."
- "Bear in mind that this doesn’t account for glare — if they replicated this test with participants trying to read in direct sunlight, there would be much more eye strain required for the LCD screens."
- "Today's screens are definitely less tiring to look at than older displays, which refreshed the image much less frequently, causing a flicker. Carl Taussig, director of Hewlett-Packard‘s Information Surfaces Lab, said the 120 Hz refresh rate typical of modern screens is much quicker than our eyes can even see.
"The new LCDs don't affect your eyes," Mr. Taussig said. "Today's screens update every eight milliseconds, whereas the human eye is moving at a speed between 10 and 30 milliseconds."
- "Glare and visual "noise": Most tablets (e.g. iPad) have a glossy screen. I believe most e-ink screens are optimized for a matte screen, which allows our eyes to relax in a more natural state, but does not address some of the other issues associated with eye strain, such as distance and reduced blink rate."
- "Quality of light: Our eyes are designed to process natural, reflected light better than self-illuminated objects (i.e. artificial light). E-Ink wins in this regard and can help reduce strain when compared to LCD screens."
- "Tablets like the iPad are much harder to read in bright light, and their screens can produce a lot of glare. E-Ink screens, on the other hand, work fantastically in bright light.
E-Ink has a low contrast ratio that is difficult to read in low light. Tablets, being backlit, are much easier to read in darker situations.
Tablets may be great in low light, but if you're reading them in the dark—particularly at night—the blue light can cause insomnia and other problems. In that case, a paper book might be best—it's easier to read than E-Ink in low light, and doesn't come with the problems of a tablet."
- "If you're looking for an ereader to accompany you to the beach, you'll definitely want to go E-Ink. If you're looking to read inside and during the daytime, an iPad or Kindle Fire may be better. And, no matter what you're reading on, take a break every 20 minutes or so if your eyes are feeling tired. That's going to be a much bigger cause of eyestrain than the type of screen you're using."
-"One driving reason for this is that they are bright. While this might makes the colors pop, it starts to hurt the eyes after prolonged use. Although many users complain of eye strain, dry eyes and headaches, no extensive scientific research has been done to prove that these are the direct effects of LCD display."
- "Light is light. There is no difference between reflected light and direct light. None. What is different is that reflected light will be in balance with the room light (mostly).
Turn your monitor light to match the room light and the strain on your eyes, for most people, goes away.
Reading on a computer screen has other issues as well. Length your eye travels to follow text across the screen, too small font size, uncomfortable posture etc."
- "When we asked William Lynch, Barnes & Noble's CEO, about the potential for eyestrain with Nook Color screen, he said the company had done extensive research on displays and discovered that eyestrain with LCDs was not the huge issue many people were making it out to be. Furthermore, the company is also using a high-resolution next-generation panel from LG that's backlit with LED."
- "A backlit or nonbacklit display doesn't make a difference, Hornfeld says. And if you're reading a bright screen in the dark, your eyes will adjust. Your pupil gets large in the dark, so when you turn on a brightly lit display, it may bother your eyes at first, but they'll compensate. It's like when you wake up in the morning, open the shades, and are blinded by the light at first. But then you get used to it."
- "Hornfeld notes that when you read or watch a movie, you simply don't blink as much, so you're eyes can get dry--especially if you're already prone to having an underlying dry-eye problem. (A New York Times article earlier this year also made a similar point about not enough moisture because of lack of blinking. It also offers some reading tactics to reduce the potential for eyestrain)."
- "Hornfeld says that today's LCD screens aren't going to give you eyestrain. That said, some people simply like the way e-ink appears on the page, and some prefer how the iPad displays text. It's an aesthetic issue more than anything else. In other words, you can simply be averse to one screen or another--but that doesn't mean it will give you eyestrain."
- "I haven't noted any difference in eyestrain from when I read on the iPad or when I read on the Kindle or Nook for long periods. Yes, the battery life is worse on an LCD device like the Nook Color than an e-ink Nook Wi-Fi or 3G. But my eyes feel about the same regardless of whether I'm using an iPad or the Kindle. The delayed flash of a e-ink page turn and sometimes ghosting of letters or images on e-ink e-readers didn't bother me as much as it does some people I know, though the latest generation e-ink readers (Kindle, Sony) have a new "Pearl" screen that helps remedy these little, nagging flaws."
- "Sure, I much prefer using an e-ink device outdoors in sunlight. Given the choice indoors, I don't have a real preference; they both look pretty good to me, and I even don't mind reading on my iPhone 3GS or iPod Touch 4G (the Retina Display seems quite good for reading, I just wish it was on my iPad). "
- "The objective tests resulted in a mix data points, with some subjects scoring higher after reading on E-ink and others scoring higher after reading on LCD. The results from the visual fatigue query were fairly close, though the LCD did rate marginally higher. Even the reading speed was virtually identical. In fact, the only major difference was when the test subjects were queried about general fatigue. LCD test subjects reported a higher general fatigue level than did E-ink test subjects, but that could be due to the size and weight of the iPad.
E-ink is often described as being just like reading on paper, and that's why the scientists were surprised to discover that the results were so similar. They were also somewhat surprised because this study contradicted many of the studies listed in the bibliography. It has long been documented that reading on screens is more tiring than reading on paper, and the bibliography cites any number of papers from the past 30 years which prove this is true.
So yes, all those assumptions about reading on screens being bad for you had a basis in fact. But here's the catch: many of those older papers might not be relevant to how we read in 2012. As the authors point out in the paper, screen technology has improved a lot in the past decade, even more so in the past few years.
What if the reason this study couldn't find a difference in was that the subjects read on an iPad? Even today, that is a high quality screen. What if the subjects in those older studies were affected more by the poor quality of the tech than the tech itself?
If this pans out then it means the screen fetishists were right all along. A higher resolution screen really is better, and each step up in resolution is a boost in the user experience. It's not just marketing hype, anymore; pixels do matter."
"There may be many reasons for eyestrain like color of display and brightness of display also. If color is reason for eyestrain then definitely E ink is more effective compared to IPad's retina display. For brightness it depends on the settings."
- "For example, the ergonomics of reading screens and the lack of blinking when we stare at them play a big role in eye fatigue. “The current problem with reading on screens is that we need to adjust our bodies to our computer screens, rather than the screens adjusting to us,” Dr. Meredith said."
- "“It depends on the viewing circumstances, including the software and typography on the screen,” said Mr. Bove. “Right now E Ink is great in sunlight, but in certain situations, a piece of paper can be a better display than E Ink, and in dim light, an LCD display can be better than all of these technologies.”"
- "E Ink has a very low contrast ratio. Although it can offer an excellent reading experience in bright sunlight, the screens can become uncomfortable to use in dark settings because of the lack of contrast and backlighting on the screen."
- "LCD screens, meanwhile, have long struggled to offer good viewing angles for reading. Apple’s latest IPS LCD screens include extremely wide viewing angles, but the reflective glass on the screen could be a hindrance in brightly lit situations."
- "Professor Alan Hedge, director of the Human Factors and Ergonomics Laboratory at Cornell University, said that reducing eye fatigue is less a matter of choosing a specific display than of taking short breaks from looking at the screen.
When we read, Dr. Hedge explained, a series of ocular muscles jump around and can cause strain, regardless of whether we are looking at pixels or paper. “While you’re reading, your eyes make about 10,000 movements an hour. It’s important to take a step back every 20 minutes and let your eyes rest,” he said.
Today’s screens are definitely less tiring to look at than older displays, which refreshed the image much less frequently, causing a flicker. Carl Taussig, director of Hewlett-Packard‘s Information Surfaces Lab, said the 120 Hz refresh rate typical of modern screens is much quicker than our eyes can even see.""
- "“The new LCDs don’t affect your eyes,” Mr. Taussig said. “Today’s screens update every eight milliseconds, whereas the human eye is moving at a speed between 10 and 30 milliseconds.”"
- "RESULTS: Results suggested that reading on the two display types is very similar in terms of both subjective and objective measures.
CONCLUSIONS: It is not the technology itself, but rather the image quality that seems crucial for reading. Compared to the visual display units used in the previous few decades, these more recent electronic displays allow for good and comfortable reading, even for extended periods of time."
- "E Ink displays are referred to as “reflective displays.” In an LCD, or “emissive display”, light from a backlight is projected through the display towards your eyes. In an E Ink display, no backlight is used; rather, ambient light from the environment is reflected from the surface of the display back to your eyes. As with any reflective surface, the more ambient light, the brighter the display looks. This attribute mimics traditional ink and paper, and users of E Ink displays have said that they do not have the same eye fatigue as with LCDs when reading for long periods of time."
- "There have been a few studies that are worth taking a look at, that talk about the benefits of e-paper vs the traditional LCD experience. The first one actually looks at an Ipad vs paper. Unsurprisingly they showed significant effects on a variety of sleep and wakefulness parameters. The next study isn’t perfect either, but it shows that the wavelengths emitted and intensity of light between an Ipad, Iphone, and Paperwhite were all very similar, but the core technology is very different."
- "Eyestrain, or eye fatigue from LCD Screens is caused by two aspects of the screen. One is the backlighting which is either supplied by CFL flourescent bulbs or recently the much brighter LED backlighting. The second is the screen refresh rate of the LCD display. The refresh rate is fatigueing to the eye due to the constant flickering of the screen as it redraws itself 60 or more times a second."
- "So I had not intent on doing research on commenting on this, but the other guy being so absolutely cock-sure of the kindle light being magically better made me want to look it up. I came across two different papers that touch on different aspects, mostly regarding sleep however. I didn't find any perfect studies fitting what you wanted, but hey it's something. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4313820/#s7 Should be available without a paywall. This one actually looks at an Ipad vs paper. Unsurprisingly they showed significant effects on a variety of sleep and wakefulness parameters. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4602096/ This next study isn't perfect either, but it shows that the wavelengths emitted and intensity of light between an Ipad, Iphone, and Paperwhite were all very similar. The rest of the paper focuses pretty exclusively on preventing these wavelengths so not very specific. Sorry, I couldn't really find anything specifically looking at the e-ink vs LCD part, but maybe this gives a little help?"
- "One advantage that the reflective displays would have compared to the emissive displays is that you can turn the built-in light off, and use the environment light as you would with a paper book. That means, theoretically, an e-ink display would not have an adverse effect on sleep like an LCD would."
- "https://www.ncbi.nlm.nih.gov/pubmed/24386252 (from 2013) talks about LCD vs E-ink, and determines backlit LCD induces eyestrain (as measured by blinks per second and self reported visual fatigue) at a higher level, and that the E-ink Paperwhite has a profile similar to paper. However, it does suffer from some of the same issues you mention (screen size, weight) and was a fairly small study."
- "Using the PRS-700BC on April 19th I found I was easily able to read more than an hour of text content without experiencing significant eyestrain. With the iPad, I was only able to go for about 45 minutes using varied brightness settings on both iBooks and Kindle for iPad.
On April 20, Sandi fared slightly better on the living room couch, citing about an hour of total tolerance to eyestrain on the iPad while adjusting various settings for text size and brightness level. Additionally, Sandra felt she would be able to read for at least 2 hours using her Kindle 2 under this lighting condition."
- "The iPad appears to be adequate for light daytime indoor reading, but fails miserably as an outdoor reading device. Vizplex e-Ink readers such as the Kindle, the SONY line and the Barnes & Noble Nook still appear to be much more optimal for long duration reading indoors and outdoors during daytime hours.
However, both Sandi and I were very surprised how well the iPad performs in dimly lit rooms and in complete darkness. If it weren't for the considerable heft of the iPad, she would strongly consider using one to replace her Kindle for reading in bed at night, citing less chance for disruption of her spouse and less eyestrain than using a Mighty Brite or similar LED clip lamp with the Vizplex display on the Kindle.
We were both at somewhat of a toss up if White on Black or Black on White was the preferred color scheme for night time reading, but agreed that the Kindle reader application was superior to iBooks overall for serious readers in any light condition."
- "For my own personal use I find White on Black to be much more acceptable to me for use during the evening in dim light or dark rooms, and it also drastically reduces the amount of white light coming out of the iPad, so it is far less likely to disturb my spouse.
However, Sandi had the opposite reaction, feeling that the White on Black text caused an unacceptable amount of eyestrain. Instead, she was able to adjust the settings in iBooks using regular Black on White to approximately 10 to 15 percent brightness which produced strain-free illumination that she could read with comfortably for several hours."
The aim of the present study was to compare prolonged reading on three different supports regarding their effects on visual fatigue. Likewise Kang et al. , and Chang et al. , variables such as font size, typeface and number of words per page were not manipulated and were kept constant across the three devices for the entire reading sessions. Subjective measures (VFS) suggested that prolonged reading on the LCD (Kindle Fire HD) triggers higher visual fatigue with respect to the E-ink (Kindle Paperwhite) and the paper book. Concerning objective measures (BPS and CFF), contrasting results were found.
As to CFF, results revealed a significant drop in sensory perception after reading independently from the device, thus failing to show significant differences among the three reading supports. On the one hand, these results are in line with previous studies employing the CFF for similar purposes, which did not succeed in finding differences between paper book, E-ink and LCD  and between E-ink and LCD . On the other hand, although our experimental plan has some similarities with that of Kang et al. , where the task consisted of reading novelettes for 40 to 60 min, our results are contrasting. These authors found a significant difference between the paper book and the LCD, namely a larger CFF reduction when reading on the LCD. In the present study, we could not replicate this finding.
With regard to BPS, experimental evidence indicated that reading on the LCD leads to a larger decrease in the number of blinks, with respect to the other supports. This result is in line with a large number of studies on CVS (for a review see ), where the use of backlit displays is usually associated with a decreased frequency of blinking and an increased rate of tear evaporation, each of which contributes to dry eyes. In fact, prolonged display exposure contributes to incomplete blinking provoking tear film instability , which is one of the main factors for visual fatigue on VDT .
In contrast with previous studies, where no differences in terms of perceived visual fatigue (VFS) were found between LCD, E-ink and paper book  and LCD and E-ink , our results showed that participants felt visually fatigued only when reading on the LCD. Such a finding might be attributable to the longer reading sessions employed in our study (on average 73 min, SD 10 min), with respect to previous studies , .
Finally, results on Subjective preference suggest that participants with no experience with e-readers prefer paper books. The overall belief that digital reading media reduce the pleasure of reading could be cultural rather than cognitive . Moreover, since reading habit for paper books is normally fixed in childhood , it’s quite obvious that people prefer paper books rather than e-books.
In conclusion, our results might be imputable to the higher level of luminance emitted by the LCD (see Materials and Methods). With respect to the paper book and the E-ink, reading on the LCD reduces the size of the pupil (APS) and the frequency of eye blink (BPS), and increases the perceived visual fatigue (VFS).
Although the Kindle Fire HD adopts a last generation LCD with IPS (in-plane switching) technology, advanced polarizing filter, and anti-glare technology, the issues related to backlight technology are still present. In contrast to LCD-displays, which have been associated with impaired reading performance  and higher visual fatigue , results on E-ink displays are encouraging.
Since visual discomfort and related symptoms occurring in VDT workers have been recognized as a growing health problem , we believe that the growing spread of e-readers should be taken into account as well. Although the aim of this study was to make an up-to-date comparison of reading devices concerning their effects on visual fatigue, it should be emphasized that comparisons with previous studies, employing older display technologies, have some limits. The use of reading devices as independent variables clearly leads to device-dependent results.
Future studies will include the manipulation of the length of the reading sessions, the luminance levels of the displays, and the study of binocular vision on prolonged reading ."
- "Despite the advances in display technology, as the computer and consumer electronics worlds go digital, human beings still have a human body that responds to analog—from the sound waves we hear to the color and lights we see. So despite CRT, TFT, LCD, plasma, or other display technologies, humans still find it easier to read off of paper.That is why e-paper, digital ink and eReaders were created--to reduce the eye strain inherent in back-lit or other display technologies. In fact, digital paper works nearly opposite the standard computer screen display. First, it looks more like paper and uses actual pigments. It is not backlit or refreshed. You can read an eReader outdoors or in direct sunlight, unlike a backlit display, which loses all of its readability in direct sunlight."
- "E-ink does not need to be refreshed like backlit technology. The image stays in place until the next electrical charge (for example, when the reader turns a page.) So an eReader display can power down to zero, using less power overall than a display that constantly needs refreshing."
- "Easy on the eyes: As already mentioned, the screen technology is nearly as easy to read as paper, however, there are other advantages. Fonts can be increased or decreased. Before, only certain books, newspapers or magazines were available in large print—now you can change the font, increase its size or zoom images, for your personal reading comfort."
(updated: Now with more reputation points I could add the beforehand cut references. If you like this post, please +1 because that allows me to add good content better in the future. :) )